Exercise 3.A.4

Question

Suppose $T \in \mathcal{L}(V,W)$ and $v_1,\ldots,v_m$ is a list of vectors in $V$ such that $Tv_1,\ldots,Tv_m$ is a linearly independent list in $W$. Prove that $v_1,\ldots,v_m$ is linearly independent.

Michael You · Accepted Answer

Suppose $v_1, \dots, v_m$ is linearly dependent, then for some $v_k$ we can write it as a linear combination of the other $v_i$'s, so we have
\begin{align*}
  v_k                               & = \sum_{i
eq k} a_iv_i       	ag{$a_i$ not all $0$} \
  T(v_k)                            & = T\left(\sum_{i
eq k} a_iv_iight)                         \
  T(v_k)                            & = \sum_{i
eq k} a_iT(v_i)                            \
  \sum_{i
eq k} a_iT(v_i) - T(v_k) & = 0
\end{align*}
the last part is a contradiction because we know $Tv_i$ are linearly dependent.

Exercise 3.A.4

Answers

Comments

Exercise 3.A.4

Answers

Comments

Add answer