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$Old$ November 5th, 2009, 01:24 AM

TeeWorlves1234124 $TeeWorlves1234124 is offline$

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$Default$ Linear Independence

Suppose that ${v_1,v_2,......,v_n}$ is a linearly independent set of vectors in $R^n$ . Show that if A is an n x n nonsingular matrix, then ${Av_1,Av_2,......,Av_n}$ is linearly independent.

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$Old$ November 5th, 2009, 04:17 AM

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Originally Posted by TeeWorlves1234124 $View Post$

Suppose that ${v_1,v_2,......,v_n}$ is a linearly independent set of vectors in $R^n$ . Show that if A is an n x n nonsingular matrix, then ${Av_1,Av_2,......,Av_n}$ is linearly independent.

For scalars $a_i$ , $0=\sum\limits_{i=1}^na_iAv_i=\sum\limits_{i=1}^nA(a_iv_i)=A\left(\sum\limits_{i=1}^na_iv_i\right)$ .

Now use that A is non-singular and then use that the given vectors are lin. indep.

Tonio

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