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Bài 6. Phụ thuộc tuyến tính và độc lập tuyến tính

by - September. 21, 2021
Kiến thức
<p><img class="wp-image-9300 aligncenter disappear appear" src="../../wp-content/uploads/2021/03/bai6-300x300.png" sizes="(max-width: 400px) 100vw, 400px" srcset="https://old.tek4.vn/wp-content/uploads/2021/03/bai6-300x300.png 300w, https://old.tek4.vn/wp-content/uploads/2021/03/bai6-150x150.png 150w, https://old.tek4.vn/wp-content/uploads/2021/03/bai6-768x768.png 768w, https://old.tek4.vn/wp-content/uploads/2021/03/bai6.png 1000w" alt="Độc lập tuyến t&igrave;nh v&agrave; phụ thuộc tuyến t&iacute;nh" width="400" height="400" loading="lazy" /></p> <p>Ở b&agrave;i trước ta đ&atilde; biết được thế n&agrave;o l&agrave; một kh&ocirc;ng gian vec tơ v&agrave; trong kh&ocirc;ng gian vec tơ được định nghĩa như thế n&agrave;o, ta cũng biết được hệ sinh của một kh&ocirc;ng gian vec tơ l&agrave; như n&agrave;o. Ở b&agrave;i n&agrave;y ta sẽ đi t&igrave;m hiểu về thế n&agrave;o l&agrave; một cơ sở kh&ocirc;ng gian, như n&agrave;o th&igrave; gọi l&agrave; vec tơ độc lập tuyến t&iacute;nh v&agrave; khi n&agrave;o l&agrave; phụ thuộc tuyến t&iacute;nh.</p> <h3 id="ftoc-heading-1" class="ftwp-heading">6.1 Phụ thuộc tuyến t&iacute;nh v&agrave; độc lập tuyến t&iacute;nh</h3> <p><strong>Định nghĩa:</strong>&nbsp;<span id="MathJax-Element-1-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-1" class="math"><span id="MathJax-Span-2" class="mrow"><span id="MathJax-Span-3" class="mi">V</span></span></span></span>&nbsp;l&agrave; một kh&ocirc;ng gian vec tơ,&nbsp;<span id="MathJax-Element-2-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-4" class="math"><span id="MathJax-Span-5" class="mrow"><span id="MathJax-Span-6" class="mi">S</span><span id="MathJax-Span-7" class="mo">=</span><span id="MathJax-Span-8" class="texatom"><span id="MathJax-Span-9" class="mrow"><span id="MathJax-Span-10" class="msubsup"><span id="MathJax-Span-11" class="mi">x</span><span id="MathJax-Span-12" class="mn">1</span></span><span id="MathJax-Span-13" class="mo">,</span><span id="MathJax-Span-14" class="mo">⋯</span><span id="MathJax-Span-15" class="mo">,</span><span id="MathJax-Span-16" class="msubsup"><span id="MathJax-Span-17" class="mi">x</span><span id="MathJax-Span-18" class="mi">n</span></span></span></span><span id="MathJax-Span-19" class="mo">&sub;</span><span id="MathJax-Span-20" class="mi">V</span></span></span></span>, ta x&eacute;t điều kiện:</p> <div class="MathJax_Display"><span id="MathJax-Element-3-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-21" class="math"><span id="MathJax-Span-22" class="mrow"><span id="MathJax-Span-23" class="msubsup"><span id="MathJax-Span-24" class="mi">c</span><span id="MathJax-Span-25" class="mn">1</span></span><span id="MathJax-Span-26" class="msubsup"><span id="MathJax-Span-27" class="mi">x</span><span id="MathJax-Span-28" class="mn">1</span></span><span id="MathJax-Span-29" class="mo">+</span><span id="MathJax-Span-30" class="mo">⋯</span><span id="MathJax-Span-31" class="mo">+</span><span id="MathJax-Span-32" class="msubsup"><span id="MathJax-Span-33" class="mi">c</span><span id="MathJax-Span-34" class="mi">n</span></span><span id="MathJax-Span-35" class="msubsup"><span id="MathJax-Span-36" class="mi">x</span><span id="MathJax-Span-37" class="mi">n</span></span><span id="MathJax-Span-38" class="mo">=</span><span id="MathJax-Span-39" class="mn">0</span><span id="MathJax-Span-40" class="mtext">&nbsp;</span><span id="MathJax-Span-41" class="mtext">&nbsp;</span><span id="MathJax-Span-42" class="mtext">&nbsp;</span><span id="MathJax-Span-43" class="mtext">&nbsp;</span><span id="MathJax-Span-44" class="mtext">&nbsp;</span><span id="MathJax-Span-45" class="mo">(</span><span id="MathJax-Span-46" class="mn">6.2.1</span><span id="MathJax-Span-47" class="mo">)</span></span></span></span></div> <p>Nếu (6.2.1) chỉ xảy ra khi&nbsp;<span id="MathJax-Element-4-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-48" class="math"><span id="MathJax-Span-49" class="mrow"><span id="MathJax-Span-50" class="msubsup"><span id="MathJax-Span-51" class="mi">c</span><span id="MathJax-Span-52" class="mn">1</span></span><span id="MathJax-Span-53" class="mo">=</span><span id="MathJax-Span-54" class="mn">0</span><span id="MathJax-Span-55" class="mo">,</span><span id="MathJax-Span-56" class="mo">⋯</span><span id="MathJax-Span-57" class="mo">,</span><span id="MathJax-Span-58" class="msubsup"><span id="MathJax-Span-59" class="mi">c</span><span id="MathJax-Span-60" class="mi">n</span></span><span id="MathJax-Span-61" class="mo">=</span><span id="MathJax-Span-62" class="mn">0</span></span></span></span>&nbsp;th&igrave; ta n&oacute;i họ&nbsp;<span id="MathJax-Element-5-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-63" class="math"><span id="MathJax-Span-64" class="mrow"><span id="MathJax-Span-65" class="mi">S</span></span></span></span>&nbsp;độc lập tuyến t&iacute;nh</p> <p>Nếu tồn t&agrave;i c&aacute;c số thực&nbsp;<span id="MathJax-Element-6-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-66" class="math"><span id="MathJax-Span-67" class="mrow"><span id="MathJax-Span-68" class="msubsup"><span id="MathJax-Span-69" class="mi">c</span><span id="MathJax-Span-70" class="mn">1</span></span><span id="MathJax-Span-71" class="mo">,</span><span id="MathJax-Span-72" class="mo">⋯</span><span id="MathJax-Span-73" class="mo">,</span><span id="MathJax-Span-74" class="msubsup"><span id="MathJax-Span-75" class="mi">c</span><span id="MathJax-Span-76" class="mi">n</span></span></span></span></span>&nbsp;kh&ocirc;ng đồng thời bằng&nbsp;<span id="MathJax-Element-7-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-77" class="math"><span id="MathJax-Span-78" class="mrow"><span id="MathJax-Span-79" class="mn">0</span></span></span></span>&nbsp;để (6.2.1) thỏa m&atilde;n th&igrave; ta n&oacute;i họ&nbsp;<span id="MathJax-Element-8-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-80" class="math"><span id="MathJax-Span-81" class="mrow"><span id="MathJax-Span-82" class="mi">S</span></span></span></span>&nbsp;phụ thuộc tuyến t&iacute;nh</p> <p>Ta sẽ xem v&iacute; dụ sau để hiểu r&otilde; hơn:</p> <p>X&eacute;t ba vec tơ&nbsp;<span id="MathJax-Element-9-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-83" class="math"><span id="MathJax-Span-84" class="mrow"><span id="MathJax-Span-85" class="msubsup"><span id="MathJax-Span-86" class="mi">x</span><span id="MathJax-Span-87" class="mn">1</span></span><span id="MathJax-Span-88" class="mo">,</span><span id="MathJax-Span-89" class="msubsup"><span id="MathJax-Span-90" class="mi">x</span><span id="MathJax-Span-91" class="mn">2</span></span><span id="MathJax-Span-92" class="mo">,</span><span id="MathJax-Span-93" class="msubsup"><span id="MathJax-Span-94" class="mi">x</span><span id="MathJax-Span-95" class="mn">3</span></span><span id="MathJax-Span-96" class="mo">&isin;</span><span id="MathJax-Span-97" class="msubsup"><span id="MathJax-Span-98" class="texatom"><span id="MathJax-Span-99" class="mrow"><span id="MathJax-Span-100" class="mi">R</span></span></span><span id="MathJax-Span-101" class="mn">4</span></span></span></span></span>:</p> <div class="MathJax_Display"><span id="MathJax-Element-10-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-102" class="math"><span id="MathJax-Span-103" class="mrow"><span id="MathJax-Span-104" class="msubsup"><span id="MathJax-Span-105" class="mi">x</span><span id="MathJax-Span-106" class="mn">1</span></span><span id="MathJax-Span-107" class="mo">=</span><span id="MathJax-Span-108" class="mrow"><span id="MathJax-Span-109" class="mo">⎡⎣⎢⎢⎢</span><span id="MathJax-Span-110" class="mtable"><span id="MathJax-Span-111" class="mtd"><span id="MathJax-Span-112" class="mrow"><span id="MathJax-Span-113" class="mn">1</span></span></span><span id="MathJax-Span-114" class="mtd"><span id="MathJax-Span-115" class="mrow"><span id="MathJax-Span-116" class="mn">2</span></span></span><span id="MathJax-Span-117" class="mtd"><span id="MathJax-Span-118" class="mrow"><span id="MathJax-Span-119" class="mo">&minus;</span><span id="MathJax-Span-120" class="mn">3</span></span></span><span id="MathJax-Span-121" class="mtd"><span id="MathJax-Span-122" class="mrow"><span id="MathJax-Span-123" class="mn">4</span></span></span></span><span id="MathJax-Span-124" class="mo">⎤⎦⎥⎥⎥</span></span><span id="MathJax-Span-125" class="mo">,</span><span id="MathJax-Span-126" class="msubsup"><span id="MathJax-Span-127" class="mi">x</span><span id="MathJax-Span-128" class="mn">2</span></span><span id="MathJax-Span-129" class="mo">=</span><span id="MathJax-Span-130" class="mrow"><span id="MathJax-Span-131" class="mo">⎡⎣⎢⎢⎢</span><span id="MathJax-Span-132" class="mtable"><span id="MathJax-Span-133" class="mtd"><span id="MathJax-Span-134" class="mrow"><span id="MathJax-Span-135" class="mn">1</span></span></span><span id="MathJax-Span-136" class="mtd"><span id="MathJax-Span-137" class="mrow"><span id="MathJax-Span-138" class="mn">1</span></span></span><span id="MathJax-Span-139" class="mtd"><span id="MathJax-Span-140" class="mrow"><span id="MathJax-Span-141" class="mn">0</span></span></span><span id="MathJax-Span-142" class="mtd"><span id="MathJax-Span-143" class="mrow"><span id="MathJax-Span-144" class="mn">2</span></span></span></span><span id="MathJax-Span-145" class="mo">⎤⎦⎥⎥⎥</span></span><span id="MathJax-Span-146" class="mo">,</span><span id="MathJax-Span-147" class="msubsup"><span id="MathJax-Span-148" class="mi">x</span><span id="MathJax-Span-149" class="mn">3</span></span><span id="MathJax-Span-150" class="mo">=</span><span id="MathJax-Span-151" class="mrow"><span id="MathJax-Span-152" class="mo">⎡⎣⎢⎢⎢</span><span id="MathJax-Span-153" class="mtable"><span id="MathJax-Span-154" class="mtd"><span id="MathJax-Span-155" class="mrow"><span id="MathJax-Span-156" class="mo">&minus;</span><span id="MathJax-Span-157" class="mn">1</span></span></span><span id="MathJax-Span-158" class="mtd"><span id="MathJax-Span-159" class="mrow"><span id="MathJax-Span-160" class="mo">&minus;</span><span id="MathJax-Span-161" class="mn">2</span></span></span><span id="MathJax-Span-162" class="mtd"><span id="MathJax-Span-163" class="mrow"><span id="MathJax-Span-164" class="mn">1</span></span></span><span id="MathJax-Span-165" class="mtd"><span id="MathJax-Span-166" class="mrow"><span id="MathJax-Span-167" class="mn">1</span></span></span></span><span id="MathJax-Span-168" class="mo">⎤⎦⎥⎥⎥</span></span></span></span></span></div> <p>Việc của ta l&agrave; đi t&igrave;m&nbsp;<span id="MathJax-Element-11-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-169" class="math"><span id="MathJax-Span-170" class="mrow"><span id="MathJax-Span-171" class="msubsup"><span id="MathJax-Span-172" class="mi">c</span><span id="MathJax-Span-173" class="mn">1</span></span><span id="MathJax-Span-174" class="mo">,</span><span id="MathJax-Span-175" class="msubsup"><span id="MathJax-Span-176" class="mi">c</span><span id="MathJax-Span-177" class="mn">2</span></span><span id="MathJax-Span-178" class="mo">,</span><span id="MathJax-Span-179" class="msubsup"><span id="MathJax-Span-180" class="mi">c</span><span id="MathJax-Span-181" class="mn">3</span></span></span></span></span>&nbsp;trong phương tr&igrave;nh:</p> <div class="MathJax_Display"><span id="MathJax-Element-12-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-182" class="math"><span id="MathJax-Span-183" class="mrow"><span id="MathJax-Span-184" class="msubsup"><span id="MathJax-Span-185" class="mi">c</span><span id="MathJax-Span-186" class="mn">1</span></span><span id="MathJax-Span-187" class="msubsup"><span id="MathJax-Span-188" class="mi">x</span><span id="MathJax-Span-189" class="mn">1</span></span><span id="MathJax-Span-190" class="mo">+</span><span id="MathJax-Span-191" class="msubsup"><span id="MathJax-Span-192" class="mi">c</span><span id="MathJax-Span-193" class="mn">2</span></span><span id="MathJax-Span-194" class="msubsup"><span id="MathJax-Span-195" class="mi">x</span><span id="MathJax-Span-196" class="mn">3</span></span><span id="MathJax-Span-197" class="mo">+</span><span id="MathJax-Span-198" class="msubsup"><span id="MathJax-Span-199" class="mi">c</span><span id="MathJax-Span-200" class="mn">3</span></span><span id="MathJax-Span-201" class="msubsup"><span id="MathJax-Span-202" class="mi">x</span><span id="MathJax-Span-203" class="mn">3</span></span><span id="MathJax-Span-204" class="mo">=</span><span id="MathJax-Span-205" class="msubsup"><span id="MathJax-Span-206" class="mi">c</span><span id="MathJax-Span-207" class="mn">1</span></span><span id="MathJax-Span-208" class="mrow"><span id="MathJax-Span-209" class="mo">⎡⎣⎢⎢⎢</span><span id="MathJax-Span-210" class="mtable"><span id="MathJax-Span-211" class="mtd"><span id="MathJax-Span-212" class="mrow"><span id="MathJax-Span-213" class="mn">1</span></span></span><span id="MathJax-Span-214" class="mtd"><span id="MathJax-Span-215" class="mrow"><span id="MathJax-Span-216" class="mn">2</span></span></span><span id="MathJax-Span-217" class="mtd"><span id="MathJax-Span-218" class="mrow"><span id="MathJax-Span-219" class="mo">&minus;</span><span id="MathJax-Span-220" class="mn">3</span></span></span><span id="MathJax-Span-221" class="mtd"><span id="MathJax-Span-222" class="mrow"><span id="MathJax-Span-223" class="mn">4</span></span></span></span><span id="MathJax-Span-224" class="mo">⎤⎦⎥⎥⎥</span></span><span id="MathJax-Span-225" class="mo">+</span><span id="MathJax-Span-226" class="msubsup"><span id="MathJax-Span-227" class="mi">c</span><span id="MathJax-Span-228" class="mn">2</span></span><span id="MathJax-Span-229" class="mrow"><span id="MathJax-Span-230" class="mo">⎡⎣⎢⎢⎢</span><span id="MathJax-Span-231" class="mtable"><span id="MathJax-Span-232" class="mtd"><span id="MathJax-Span-233" class="mrow"><span id="MathJax-Span-234" class="mn">1</span></span></span><span id="MathJax-Span-235" class="mtd"><span id="MathJax-Span-236" class="mrow"><span id="MathJax-Span-237" class="mn">1</span></span></span><span id="MathJax-Span-238" class="mtd"><span id="MathJax-Span-239" class="mrow"><span id="MathJax-Span-240" class="mn">0</span></span></span><span id="MathJax-Span-241" class="mtd"><span id="MathJax-Span-242" class="mrow"><span id="MathJax-Span-243" class="mn">2</span></span></span></span><span id="MathJax-Span-244" class="mo">⎤⎦⎥⎥⎥</span></span><span id="MathJax-Span-245" class="mo">+</span><span id="MathJax-Span-246" class="msubsup"><span id="MathJax-Span-247" class="mi">c</span><span id="MathJax-Span-248" class="mn">3</span></span><span id="MathJax-Span-249" class="mrow"><span id="MathJax-Span-250" class="mo">⎡⎣⎢⎢⎢</span><span id="MathJax-Span-251" class="mtable"><span id="MathJax-Span-252" class="mtd"><span id="MathJax-Span-253" class="mrow"><span id="MathJax-Span-254" class="mo">&minus;</span><span id="MathJax-Span-255" class="mn">1</span></span></span><span id="MathJax-Span-256" class="mtd"><span id="MathJax-Span-257" class="mrow"><span id="MathJax-Span-258" class="mo">&minus;</span><span id="MathJax-Span-259" class="mn">2</span></span></span><span id="MathJax-Span-260" class="mtd"><span id="MathJax-Span-261" class="mrow"><span id="MathJax-Span-262" class="mn">1</span></span></span><span id="MathJax-Span-263" class="mtd"><span id="MathJax-Span-264" class="mrow"><span id="MathJax-Span-265" class="mn">1</span></span></span></span><span id="MathJax-Span-266" class="mo">⎤⎦⎥⎥⎥</span></span><span id="MathJax-Span-267" class="mo">=</span><span id="MathJax-Span-268" class="mn">0</span></span></span></span></div> <p>hay:</p> <div class="MathJax_Display"><span id="MathJax-Element-13-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-269" class="math"><span id="MathJax-Span-270" class="mrow"><span id="MathJax-Span-271" class="msubsup"><span id="MathJax-Span-272" class="mi">c</span><span id="MathJax-Span-273" class="mn">1</span></span><span id="MathJax-Span-274" class="mrow"><span id="MathJax-Span-275" class="mo">⎡⎣⎢⎢⎢</span><span id="MathJax-Span-276" class="mtable"><span id="MathJax-Span-277" class="mtd"><span id="MathJax-Span-278" class="mrow"><span id="MathJax-Span-279" class="mn">1</span></span></span><span id="MathJax-Span-280" class="mtd"><span id="MathJax-Span-281" class="mrow"><span id="MathJax-Span-282" class="mn">2</span></span></span><span id="MathJax-Span-283" class="mtd"><span id="MathJax-Span-284" class="mrow"><span id="MathJax-Span-285" class="mo">&minus;</span><span id="MathJax-Span-286" class="mn">3</span></span></span><span id="MathJax-Span-287" class="mtd"><span id="MathJax-Span-288" class="mrow"><span id="MathJax-Span-289" class="mn">4</span></span></span></span><span id="MathJax-Span-290" class="mo">⎤⎦⎥⎥⎥</span></span><span id="MathJax-Span-291" class="mo">+</span><span id="MathJax-Span-292" class="msubsup"><span id="MathJax-Span-293" class="mi">c</span><span id="MathJax-Span-294" class="mn">2</span></span><span id="MathJax-Span-295" class="mrow"><span id="MathJax-Span-296" class="mo">⎡⎣⎢⎢⎢</span><span id="MathJax-Span-297" class="mtable"><span id="MathJax-Span-298" class="mtd"><span id="MathJax-Span-299" class="mrow"><span id="MathJax-Span-300" class="mn">1</span></span></span><span id="MathJax-Span-301" class="mtd"><span id="MathJax-Span-302" class="mrow"><span id="MathJax-Span-303" class="mn">1</span></span></span><span id="MathJax-Span-304" class="mtd"><span id="MathJax-Span-305" class="mrow"><span id="MathJax-Span-306" class="mn">0</span></span></span><span id="MathJax-Span-307" class="mtd"><span id="MathJax-Span-308" class="mrow"><span id="MathJax-Span-309" class="mn">2</span></span></span></span><span id="MathJax-Span-310" class="mo">⎤⎦⎥⎥⎥</span></span><span id="MathJax-Span-311" class="mo">+</span><span id="MathJax-Span-312" class="msubsup"><span id="MathJax-Span-313" class="mi">c</span><span id="MathJax-Span-314" class="mn">3</span></span><span id="MathJax-Span-315" class="mrow"><span id="MathJax-Span-316" class="mo">⎡⎣⎢⎢⎢</span><span id="MathJax-Span-317" class="mtable"><span id="MathJax-Span-318" class="mtd"><span id="MathJax-Span-319" class="mrow"><span id="MathJax-Span-320" class="mo">&minus;</span><span id="MathJax-Span-321" class="mn">1</span></span></span><span id="MathJax-Span-322" class="mtd"><span id="MathJax-Span-323" class="mrow"><span id="MathJax-Span-324" class="mo">&minus;</span><span id="MathJax-Span-325" class="mn">2</span></span></span><span id="MathJax-Span-326" class="mtd"><span id="MathJax-Span-327" class="mrow"><span id="MathJax-Span-328" class="mn">1</span></span></span><span id="MathJax-Span-329" class="mtd"><span id="MathJax-Span-330" class="mrow"><span id="MathJax-Span-331" class="mn">1</span></span></span></span><span id="MathJax-Span-332" class="mo">⎤⎦⎥⎥⎥</span></span><span id="MathJax-Span-333" class="mo">=</span><span id="MathJax-Span-334" class="mrow"><span id="MathJax-Span-335" class="mo">⎡⎣⎢⎢⎢</span><span id="MathJax-Span-336" class="mtable"><span id="MathJax-Span-337" class="mtd"><span id="MathJax-Span-338" class="mrow"><span id="MathJax-Span-339" class="mn">0</span></span></span><span id="MathJax-Span-340" class="mtd"><span id="MathJax-Span-341" class="mrow"><span id="MathJax-Span-342" class="mn">0</span></span></span><span id="MathJax-Span-343" class="mtd"><span id="MathJax-Span-344" class="mrow"><span id="MathJax-Span-345" class="mn">0</span></span></span><span id="MathJax-Span-346" class="mtd"><span id="MathJax-Span-347" class="mrow"><span id="MathJax-Span-348" class="mn">0</span></span></span></span><span id="MathJax-Span-349" class="mo">⎤⎦⎥⎥⎥</span></span><span id="MathJax-Span-350" class="mo">(</span><span id="MathJax-Span-351" class="mn">6.1.2</span><span id="MathJax-Span-352" class="mo">)</span></span></span></span></div> <p>Ta sẽ đi biến đổi ma trận&nbsp;<span id="MathJax-Element-14-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-353" class="math"><span id="MathJax-Span-354" class="mrow"><span id="MathJax-Span-355" class="mrow"><span id="MathJax-Span-356" class="mo">⎡⎣⎢⎢⎢⎢</span><span id="MathJax-Span-357" class="mtable"><span id="MathJax-Span-358" class="mtd"><span id="MathJax-Span-359" class="mrow"><span id="MathJax-Span-360" class="mn">1</span></span></span><span id="MathJax-Span-376" class="mtd"><span id="MathJax-Span-377" class="mrow"><span id="MathJax-Span-378" class="mn">2</span></span></span><span id="MathJax-Span-394" class="mtd"><span id="MathJax-Span-395" class="mrow"><span id="MathJax-Span-396" class="mo">&minus;</span><span id="MathJax-Span-397" class="mn">3</span></span></span><span id="MathJax-Span-412" class="mtd"><span id="MathJax-Span-413" class="mrow"><span id="MathJax-Span-414" class="mn">4</span></span></span><span id="MathJax-Span-361" class="mtd"><span id="MathJax-Span-362" class="mrow"><span id="MathJax-Span-363" class="mn">1</span></span></span><span id="MathJax-Span-379" class="mtd"><span id="MathJax-Span-380" class="mrow"><span id="MathJax-Span-381" class="mn">1</span></span></span><span id="MathJax-Span-398" class="mtd"><span id="MathJax-Span-399" class="mrow"><span id="MathJax-Span-400" class="mn">0</span></span></span><span id="MathJax-Span-415" class="mtd"><span id="MathJax-Span-416" class="mrow"><span id="MathJax-Span-417" class="mn">2</span></span></span><span id="MathJax-Span-364" class="mtd"><span id="MathJax-Span-365" class="mrow"><span id="MathJax-Span-366" class="mo">&minus;</span><span id="MathJax-Span-367" class="mn">1</span></span></span><span id="MathJax-Span-382" class="mtd"><span id="MathJax-Span-383" class="mrow"><span id="MathJax-Span-384" class="mo">&minus;</span><span id="MathJax-Span-385" class="mn">2</span></span></span><span id="MathJax-Span-401" class="mtd"><span id="MathJax-Span-402" class="mrow"><span id="MathJax-Span-403" class="mn">1</span></span></span><span id="MathJax-Span-418" class="mtd"><span id="MathJax-Span-419" class="mrow"><span id="MathJax-Span-420" class="mn">1</span></span></span><span id="MathJax-Span-368" class="mtd"><span id="MathJax-Span-369" class="mrow"><span id="MathJax-Span-370" class="texatom"><span id="MathJax-Span-371" class="mrow"><span id="MathJax-Span-372" class="mo">|</span></span></span></span></span><span id="MathJax-Span-386" class="mtd"><span id="MathJax-Span-387" class="mrow"><span id="MathJax-Span-388" class="texatom"><span id="MathJax-Span-389" class="mrow"><span id="MathJax-Span-390" class="mo">|</span></span></span></span></span><span id="MathJax-Span-404" class="mtd"><span id="MathJax-Span-405" class="mrow"><span id="MathJax-Span-406" class="texatom"><span id="MathJax-Span-407" class="mrow"><span id="MathJax-Span-408" class="mo">|</span></span></span></span></span><span id="MathJax-Span-421" class="mtd"><span id="MathJax-Span-422" class="mrow"><span id="MathJax-Span-423" class="texatom"><span id="MathJax-Span-424" class="mrow"><span id="MathJax-Span-425" class="mo">|</span></span></span></span></span><span id="MathJax-Span-373" class="mtd"><span id="MathJax-Span-374" class="mrow"><span id="MathJax-Span-375" class="mn">0</span></span></span><span id="MathJax-Span-391" class="mtd"><span id="MathJax-Span-392" class="mrow"><span id="MathJax-Span-393" class="mn">0</span></span></span><span id="MathJax-Span-409" class="mtd"><span id="MathJax-Span-410" class="mrow"><span id="MathJax-Span-411" class="mn">0</span></span></span><span id="MathJax-Span-426" class="mtd"><span id="MathJax-Span-427" class="mrow"><span id="MathJax-Span-428" class="mn">0</span></span></span></span><span id="MathJax-Span-429" class="mo">⎤⎦⎥⎥⎥⎥</span></span></span></span></span>&nbsp;về dạng ma trận bậc thang, ở đ&acirc;y m&igrave;nh sẽ viết ngắn gọn kết quả sau khi đ&atilde; qua c&aacute;c bước biến đổi sơ cấp, sau khi biến đổi sơ cấp ta được kết quả:</p> <div class="MathJax_Display"><span id="MathJax-Element-15-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-430" class="math"><span id="MathJax-Span-431" class="mrow"><span id="MathJax-Span-432" class="mrow"><span id="MathJax-Span-433" class="mo">⎡⎣⎢⎢⎢⎢</span><span id="MathJax-Span-434" class="mtable"><span id="MathJax-Span-435" class="mtd"><span id="MathJax-Span-436" class="mrow"><span id="MathJax-Span-437" class="mn">1</span></span></span><span id="MathJax-Span-453" class="mtd"><span id="MathJax-Span-454" class="mrow"><span id="MathJax-Span-455" class="mn">0</span></span></span><span id="MathJax-Span-470" class="mtd"><span id="MathJax-Span-471" class="mrow"><span id="MathJax-Span-472" class="mn">0</span></span></span><span id="MathJax-Span-487" class="mtd"><span id="MathJax-Span-488" class="mrow"><span id="MathJax-Span-489" class="mn">0</span></span></span><span id="MathJax-Span-438" class="mtd"><span id="MathJax-Span-439" class="mrow"><span id="MathJax-Span-440" class="mn">1</span></span></span><span id="MathJax-Span-456" class="mtd"><span id="MathJax-Span-457" class="mrow"><span id="MathJax-Span-458" class="mn">1</span></span></span><span id="MathJax-Span-473" class="mtd"><span id="MathJax-Span-474" class="mrow"><span id="MathJax-Span-475" class="mn">0</span></span></span><span id="MathJax-Span-490" class="mtd"><span id="MathJax-Span-491" class="mrow"><span id="MathJax-Span-492" class="mn">0</span></span></span><span id="MathJax-Span-441" class="mtd"><span id="MathJax-Span-442" class="mrow"><span id="MathJax-Span-443" class="mo">&minus;</span><span id="MathJax-Span-444" class="mn">1</span></span></span><span id="MathJax-Span-459" class="mtd"><span id="MathJax-Span-460" class="mrow"><span id="MathJax-Span-461" class="mn">0</span></span></span><span id="MathJax-Span-476" class="mtd"><span id="MathJax-Span-477" class="mrow"><span id="MathJax-Span-478" class="mn">1</span></span></span><span id="MathJax-Span-493" class="mtd"><span id="MathJax-Span-494" class="mrow"><span id="MathJax-Span-495" class="mn">0</span></span></span><span id="MathJax-Span-445" class="mtd"><span id="MathJax-Span-446" class="mrow"><span id="MathJax-Span-447" class="texatom"><span id="MathJax-Span-448" class="mrow"><span id="MathJax-Span-449" class="mo">|</span></span></span></span></span><span id="MathJax-Span-462" class="mtd"><span id="MathJax-Span-463" class="mrow"><span id="MathJax-Span-464" class="texatom"><span id="MathJax-Span-465" class="mrow"><span id="MathJax-Span-466" class="mo">|</span></span></span></span></span><span id="MathJax-Span-479" class="mtd"><span id="MathJax-Span-480" class="mrow"><span id="MathJax-Span-481" class="texatom"><span id="MathJax-Span-482" class="mrow"><span id="MathJax-Span-483" class="mo">|</span></span></span></span></span><span id="MathJax-Span-496" class="mtd"><span id="MathJax-Span-497" class="mrow"><span id="MathJax-Span-498" class="texatom"><span id="MathJax-Span-499" class="mrow"><span id="MathJax-Span-500" class="mo">|</span></span></span></span></span><span id="MathJax-Span-450" class="mtd"><span id="MathJax-Span-451" class="mrow"><span id="MathJax-Span-452" class="mn">0</span></span></span><span id="MathJax-Span-467" class="mtd"><span id="MathJax-Span-468" class="mrow"><span id="MathJax-Span-469" class="mn">0</span></span></span><span id="MathJax-Span-484" class="mtd"><span id="MathJax-Span-485" class="mrow"><span id="MathJax-Span-486" class="mn">0</span></span></span><span id="MathJax-Span-501" class="mtd"><span id="MathJax-Span-502" class="mrow"><span id="MathJax-Span-503" class="mn">0</span></span></span></span><span id="MathJax-Span-504" class="mo">⎤⎦⎥⎥⎥⎥</span></span></span></span></span></div> <p>Từ ma trận tr&ecirc;n ta dễ thấy phương tr&igrave;nh (6.1.2) c&oacute; duy nhất một nghiệm tầm thường l&agrave;&nbsp;<span id="MathJax-Element-16-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-505" class="math"><span id="MathJax-Span-506" class="mrow"><span id="MathJax-Span-507" class="msubsup"><span id="MathJax-Span-508" class="mi">c</span><span id="MathJax-Span-509" class="mn">1</span></span><span id="MathJax-Span-510" class="mo">=</span><span id="MathJax-Span-511" class="mn">0</span><span id="MathJax-Span-512" class="mo">,</span><span id="MathJax-Span-513" class="msubsup"><span id="MathJax-Span-514" class="mi">c</span><span id="MathJax-Span-515" class="mn">2</span></span><span id="MathJax-Span-516" class="mo">=</span><span id="MathJax-Span-517" class="mn">0</span><span id="MathJax-Span-518" class="mo">,</span><span id="MathJax-Span-519" class="msubsup"><span id="MathJax-Span-520" class="mi">c</span><span id="MathJax-Span-521" class="mn">3</span></span><span id="MathJax-Span-522" class="mo">=</span><span id="MathJax-Span-523" class="mn">0</span></span></span></span>&nbsp;do đ&oacute; tập hợp c&aacute;c vec tơ&nbsp;<span id="MathJax-Element-17-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-524" class="math"><span id="MathJax-Span-525" class="mrow"><span id="MathJax-Span-526" class="msubsup"><span id="MathJax-Span-527" class="mi">x</span><span id="MathJax-Span-528" class="mn">1</span></span><span id="MathJax-Span-529" class="mo">,</span><span id="MathJax-Span-530" class="msubsup"><span id="MathJax-Span-531" class="mi">x</span><span id="MathJax-Span-532" class="mn">2</span></span><span id="MathJax-Span-533" class="mo">,</span><span id="MathJax-Span-534" class="msubsup"><span id="MathJax-Span-535" class="mi">x</span><span id="MathJax-Span-536" class="mn">3</span></span></span></span></span>&nbsp;l&agrave; độc lập tuyến t&iacute;nh.</p> <h3 id="ftoc-heading-2" class="ftwp-heading">6.2 Vec tơ cơ sở</h3> <p><strong>Định nghĩa:</strong>&nbsp;Giả sử ta c&oacute; kh&ocirc;ng gian vec tơ&nbsp;<span id="MathJax-Element-18-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-537" class="math"><span id="MathJax-Span-538" class="mrow"><span id="MathJax-Span-539" class="mi">V</span></span></span></span>, v&agrave;&nbsp;<span id="MathJax-Element-19-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-540" class="math"><span id="MathJax-Span-541" class="mrow"><span id="MathJax-Span-542" class="mi">S</span><span id="MathJax-Span-543" class="mo">=</span><span id="MathJax-Span-544" class="texatom"><span id="MathJax-Span-545" class="mrow"><span id="MathJax-Span-546" class="mi">e</span><span id="MathJax-Span-547" class="mn">1</span><span id="MathJax-Span-548" class="mo">,</span><span id="MathJax-Span-549" class="mo">⋯</span><span id="MathJax-Span-550" class="mo">,</span><span id="MathJax-Span-551" class="msubsup"><span id="MathJax-Span-552" class="mi">e</span><span id="MathJax-Span-553" class="mi">n</span></span></span></span></span></span></span>&nbsp;l&agrave; một họ&nbsp;<span id="MathJax-Element-20-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-554" class="math"><span id="MathJax-Span-555" class="mrow"><span id="MathJax-Span-556" class="mi">n</span></span></span></span>&nbsp;vec tơ của&nbsp;<span id="MathJax-Element-21-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-557" class="math"><span id="MathJax-Span-558" class="mrow"><span id="MathJax-Span-559" class="mi">V</span></span></span></span>, Nếu&nbsp;<span id="MathJax-Element-22-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-560" class="math"><span id="MathJax-Span-561" class="mrow"><span id="MathJax-Span-562" class="mi">S</span></span></span></span>&nbsp;sinh ra&nbsp;<span id="MathJax-Element-23-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-563" class="math"><span id="MathJax-Span-564" class="mrow"><span id="MathJax-Span-565" class="mi">V</span></span></span></span>&nbsp;v&agrave; độc lập tuyến t&iacute;nh th&igrave;&nbsp;<span id="MathJax-Element-24-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-566" class="math"><span id="MathJax-Span-567" class="mrow"><span id="MathJax-Span-568" class="mi">V</span></span></span></span>&nbsp;l&agrave; kh&ocirc;ng gian&nbsp;<span id="MathJax-Element-25-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-569" class="math"><span id="MathJax-Span-570" class="mrow"><span id="MathJax-Span-571" class="mi">n</span></span></span></span>&nbsp;chiều v&agrave;&nbsp;<span id="MathJax-Element-26-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-572" class="math"><span id="MathJax-Span-573" class="mrow"><span id="MathJax-Span-574" class="mi">S</span></span></span></span>&nbsp;l&agrave; cơ sở của&nbsp;<span id="MathJax-Element-27-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-575" class="math"><span id="MathJax-Span-576" class="mrow"><span id="MathJax-Span-577" class="mi">V</span></span></span></span>.</p> <p>V&iacute; dụ về hệ cơ sở: Trong kh&ocirc;ng gian hai chiều&nbsp;<span id="MathJax-Element-28-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-578" class="math"><span id="MathJax-Span-579" class="mrow"><span id="MathJax-Span-580" class="msubsup"><span id="MathJax-Span-581" class="texatom"><span id="MathJax-Span-582" class="mrow"><span id="MathJax-Span-583" class="mi">R</span></span></span><span id="MathJax-Span-584" class="mn">2</span></span></span></span></span>&nbsp;ta c&oacute; hai vec tơ cơ sở l&agrave;&nbsp;<span id="MathJax-Element-29-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-585" class="math"><span id="MathJax-Span-586" class="mrow"><span id="MathJax-Span-587" class="msubsup"><span id="MathJax-Span-588" class="mi">e</span><span id="MathJax-Span-589" class="mn">1</span></span><span id="MathJax-Span-590" class="mo">(</span><span id="MathJax-Span-591" class="mn">1</span><span id="MathJax-Span-592" class="mo">;</span><span id="MathJax-Span-593" class="mn">0</span><span id="MathJax-Span-594" class="mo">)</span></span></span></span>&nbsp;v&agrave;&nbsp;<span id="MathJax-Element-30-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-595" class="math"><span id="MathJax-Span-596" class="mrow"><span id="MathJax-Span-597" class="msubsup"><span id="MathJax-Span-598" class="mi">e</span><span id="MathJax-Span-599" class="mn">2</span></span><span id="MathJax-Span-600" class="mo">(</span><span id="MathJax-Span-601" class="mn">0</span><span id="MathJax-Span-602" class="mo">;</span><span id="MathJax-Span-603" class="mn">1</span><span id="MathJax-Span-604" class="mo">)</span></span></span></span>, trong kh&ocirc;ng gian ba chiều&nbsp;<span id="MathJax-Element-31-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-605" class="math"><span id="MathJax-Span-606" class="mrow"><span id="MathJax-Span-607" class="msubsup"><span id="MathJax-Span-608" class="texatom"><span id="MathJax-Span-609" class="mrow"><span id="MathJax-Span-610" class="mi">R</span></span></span><span id="MathJax-Span-611" class="mn">3</span></span></span></span></span>&nbsp;ta c&oacute; ba vec vec tơ cơ sở l&agrave;&nbsp;<span id="MathJax-Element-32-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-612" class="math"><span id="MathJax-Span-613" class="mrow"><span id="MathJax-Span-614" class="msubsup"><span id="MathJax-Span-615" class="mi">e</span><span id="MathJax-Span-616" class="mn">1</span></span><span id="MathJax-Span-617" class="mo">(</span><span id="MathJax-Span-618" class="mn">1</span><span id="MathJax-Span-619" class="mo">;</span><span id="MathJax-Span-620" class="mn">0</span><span id="MathJax-Span-621" class="mo">;</span><span id="MathJax-Span-622" class="mn">0</span><span id="MathJax-Span-623" class="mo">)</span></span></span></span>,&nbsp;<span id="MathJax-Element-33-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-624" class="math"><span id="MathJax-Span-625" class="mrow"><span id="MathJax-Span-626" class="msubsup"><span id="MathJax-Span-627" class="mi">e</span><span id="MathJax-Span-628" class="mn">2</span></span><span id="MathJax-Span-629" class="mo">(</span><span id="MathJax-Span-630" class="mn">0</span><span id="MathJax-Span-631" class="mo">;</span><span id="MathJax-Span-632" class="mn">1</span><span id="MathJax-Span-633" class="mo">;</span><span id="MathJax-Span-634" class="mn">0</span><span id="MathJax-Span-635" class="mo">)</span></span></span></span>&nbsp;v&agrave;&nbsp;<span id="MathJax-Element-34-Frame" class="MathJax" tabindex="0"><span id="MathJax-Span-636" class="math"><span id="MathJax-Span-637" class="mrow"><span id="MathJax-Span-638" class="msubsup"><span id="MathJax-Span-639" class="mi">e</span><span id="MathJax-Span-640" class="mn">3</span></span><span id="MathJax-Span-641" class="mo">(</span><span id="MathJax-Span-642" class="mn">0</span><span id="MathJax-Span-643" class="mo">;</span><span id="MathJax-Span-644" class="mn">0</span><span id="MathJax-Span-645" class="mo">;</span><span id="MathJax-Span-646" class="mn">1</span><span id="MathJax-Span-647" class="mo">)</span></span></span></span>. C&aacute;c vec tơ m&igrave;nh vừa lấy v&iacute; dụ c&ograve;n gọi l&agrave; vec tơ đơn vị, c&aacute;c vec tơ n&agrave;y sẽ c&oacute; chiều d&agrave;i l&agrave; 1 v&agrave; vu&ocirc;ng g&oacute;c với nhau, nếu để &yacute; th&igrave; ta c&oacute; thể dễ thấy c&aacute;c hệ tọa độ m&agrave; thời phổ th&ocirc;ng ta hay d&ugrave;ng cũng đều l&agrave; kh&ocirc;ng gian c&oacute; c&aacute;c vec tơ cơ sở l&agrave; c&aacute;c vec tơ đơn vị n&agrave;y.</p> <p>C&acirc;u hỏi đặt ra l&agrave; liệu c&aacute;c vec tơ cơ sở n&agrave;y c&oacute; nhất thiết phải c&oacute; chiều d&agrave;i l&agrave; một v&agrave; vu&ocirc;ng g&oacute;c với nhau kh&ocirc;ng ? C&acirc;u trả lời l&agrave; kh&ocirc;ng, c&aacute;c vec tơ n&agrave;y c&oacute; thể kh&ocirc;ng c&oacute; chiều d&agrave;i l&agrave; một v&agrave; cũng c&oacute;&nbsp; thể kh&ocirc;ng cần vu&ocirc;ng g&oacute;c với nhau, nếu c&aacute;c vec tơ cơ sở kh&ocirc;ng vu&ocirc;ng g&oacute;c với nhau th&igrave; trục tung v&agrave; trục ho&agrave;nh cũng kh&ocirc;ng vu&ocirc;ng g&oacute;c với nhau, chẳng qua khi sử dụng vec tơ cơ sở l&agrave; c&aacute;c vec tơ đơn vị th&igrave; ta sẽ dễ d&agrave;ng thao t&aacute;c t&iacute;nh to&aacute;n hơn, do c&aacute;c vec tơ đơn vị đều vu&ocirc;ng g&oacute;c với nhau.</p>