Reversing Linear Transformations

Throughout this post, let $V,W$ be vector spaces over some field $\mathbb F$ and $T : V \to W$ be a linear transformation.

Problem 1. For any subspace $K \subseteq V$ , prove that $T(K)$ is a subspace of $W$ . In particular, the range $T(V)$ of $T$ is a subspace of $W$ .

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Solution. For any $x,y \in K$ and for any scalar $c \in \mathbb F$ ,

$T(x) + T(y) = T(x+y) \in T(K),\quad cT(x) = T(cx) \in T(K).$

Problem 2. For any subspace $L \subseteq W$ , prove that $T^{-1}(L)$ is a subspace of $V$ . In particular, the kernel $\mathrm{ker}(T):= T^{-1}(\{0\})$ of $T$ is a subspace of $V$ .

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Solution. For any $x,y \in T^{-1}(L)$ , $T(x),T(y) \in K$ . Since $K$ is a subspace of $W$ , for any scalar $c \in \mathbb F$ ,

$T(x+y) = T(x) + T(y) \in L,\quad T(cx) = cT(x) \in L.$

Problem 3. For any $x, y \in V$ , prove that $T(x) = T(y)$ if and only if $x-y \in \mathrm{ker}(T)$ .

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Solution. We have $T(x) = T(y) \iff T(x) - T(y) = 0 \iff T(x-y) = 0$ .

Problem 4. For subspaces $K_1, K_2 \subseteq W$ , prove that

$K_1 +K_2 := \{x + y : x \in K_1, y \in K_2\}$

is a subspace of $W$ . Also, for any $c \in \mathbb F$ , prove that $cK_1 := \{cx : x \in K_1\}$ is a subspace of $W$ .

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Solution. For any $x_1+y_1, x_2+y_2 \in K_1 + K_2$ ,

$(x_1+y_1) + (x_2+y_2) = (x_1 + x_2) + (y_1 + y_2) \in K_1 + K_2.$

Similarly, for any scalar $c \in \mathbb F$ ,

$c(x_1+y_1) = c x_1 + c x_2 \in K_1 + K_2.$

The proof for $cK_1$ is similar.

Problem 5. For subspaces $K_1, K_2 \subseteq W$ , prove that

$T^{-1}(K_1 + K_2) \supseteq T^{-1}(K_1) + T^{-1}(K_2) + \mathrm{ker}(T),$

where equality holds if $T$ is surjective.

(Click for Solution)

Solution. For the first claim, to prove $(\supseteq)$ , we note that for any $x_i \in T^{-1}(K_i)$ , $x_3 \in \mathrm{ker}(T)$ ,

$T(x_1+x_2+x_3) = T(x_1) + T(x_2) \in K_1 + K_2$

so that $x_1+x_2+x_3 \in T^{-1}(K_1 + K_2)$ .

To prove $(\subseteq)$ when $T$ is surjective, fix $x \in T^{-1}(K_1 + K_2)$ . Then $T(x) \in K_1 + K_2$ . Hence, there exists $y_i \in K_i$ such that $T(x) = y_1 + y_2$ .

Since $T$ is surjective, find $x_i \in V$ such that $T(x_i) = y_i \in K_i \Rightarrow x_i \in T^{-1}(K_i)$ . Then $T(x) = T(x_1) + T(x_2) = T(x_1 + x_2)$ . By Problem 3,