Solution 3a 13

Suppose $v_1,\dots,v_m$ is a linearly dependent list of vectors in $V$ . Suppose also that $W \ne \{0\}$ . Prove that there exists $w_1,\dots,w_m \in W$ such that no $T \in \mathcal L(V,W)$ satisfies $Tv_k = w_k$ for each $k=1,\dots,m$ .

Let $w \ne 0$ be a vector in $W$ . Our plan is to set $w_k = \lambda_k w$ in such a way that defining $Tv_k = w_k$ leads to a contradiction because of dependence.

By 2.21 we can find the first dependent vector $v_j$ with

v_j = a_1v_1+\dots+a_{j-1}v_{j-1}

Define $w_1,\dots,w_j$ so that

w_j = a_1w_1+\dots+a_{j-1}w_{j-1}

Then having $Tv_k = w_k$ would not be a valid linear transformation. Since using it leads to a contradiction when we apply it to both sides of

v_j = a_1 + \dots + a_{j-1}v_{j-1}

And use linearity.