1
$\begingroup$

If $L:V \rightarrow W$ is a linear function and $\{v_1,\ldots,v_n\}$ is linearly dependent, then $\{L(v_1),\ldots,L(v_n)\}$ is linearly dependent. Is this statement true?

When $\{v_1,\ldots,v_n\}$ is said to be linearly dependent, it is implying that multiple of some scalar equals to some other vector other than the zero vector, right?

Then $\{L(v_1),\ldots,L(v_n)\}$ should be linearly dependent since $\{L(v_1),\ldots,L(v_n)\}$ are nonzero vectors. Is that right..? Im not sure

$\endgroup$

1 Answer 1

1
$\begingroup$

The statement is true, but I don't think you've proven it. To see it, suppose you have a nontrivial combination of the $v_i$ which is zero. In other words you have $\sum_{i=1}^n c_i v_i = 0$ where at least one $c_i$ is nonzero. Then $L(\sum_{i=1}^n c_i v_i)=\sum_{i=1}^n c_i L(v_i)=L(0)=0$.

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .