document.write( "Question 1031968: Let L:V→W be a linear transformation. Let {(X1),(X2),…, (Xn)} ϵ V. IF {L(X1),L(X2),…, L(Xn)} is linearly dependent, then {X1,X2,...,Xn} is linearly dependent. \n" ); document.write( "

Algebra.Com's Answer #646686 by ikleyn(52803) $\"\"$ $\"About$

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\n" ); document.write( "It is not necessarily true.\r
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\n" ); document.write( "\n" ); document.write( "I just said it, and I am repeating it one more time and again.\r
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\n" ); document.write( "\n" ); document.write( ">>> IT IS NOT NECESSARILY TRUE. <<<\r
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\n" ); document.write( "\n" ); document.write( "It is true only in the case when the operator L is non-degenerated (has the zero kernel).
\n" ); document.write( "Which is not always the case for linear transformations.\r
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\n" ); document.write( "\n" ); document.write( "The proof of the other tutor is wrong, unfortunately.\r
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document.write( "A contr-example is:\r\n" );
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document.write( "Take 3 linearly independent vectors in .\r\n" );
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document.write( "Let the operator L be the projection  on .\r\n" );
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document.write( "Every three vectors in  are dependent.\r\n" );
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document.write( "So are dependent in  the projections of the original vectors from {{R^3}}}.\r\n" );
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document.write( "But the original vectors were chosen as linearly independent. \r\n" );
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document.write( "It is on the level of elementary knowledge of linear algebra.\r\n" );
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\n" ); document.write( "\n" ); document.write( "Again: the fact that the images are linearly dependent DOES NOT IMPLY that the pre-images are necessarily linearly dependent.\r
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