It looks like you've uploaded an image containing several problems related to linear algebra and vector spaces. Here’s a breakdown of the four problems presented in the image:

Problem 1:
Let $U, V, W$ be vector spaces over a field $\mathbb{F}$. Let $L(U, V)$ denote the set of linear maps from $U$ to $V$.

• (a) Show that $L(U, V)$ is a vector space. (Optional, ungraded)
• (b) Suppose $T_1: U \to V$ and $T_2: V \to W$ are linear. Prove that $T_2 \circ T_1$ is linear.
• (c) Suppose $f: U \to V$ is a linear bijection and let $f^{-1}: V \to U$ be its inverse. Prove that $f^{-1}$ is linear.

Problem 2:
Let $U$ and $V$ be vector spaces over a field $\mathbb{F}$, and let $T_1: U \to V$ and $T_2: U \to V$ be linear. Suppose $S = \{u_1, \dots, u_k\} \subset U$ and that for all $s \in S$ we have $T_1(s) = T_2(s)$. Prove that for all $x \in \text{span}(S)$ we have that $T_1(x) = T_2(x)$.

Problem 3:
Let $V$ and $W$ be vector spaces over a field $\mathbb{F}$, and let $T: V \to W$ be linear. Let $S = \{u_1, \dots, u_k\} \subset V$ and define $T(S) = \{T(u) \mid u \in S\}$.

• (a) If $S$ spans $V$, prove that $T(S)$ spans the image of $T$.
• (b) If $T(S)$ is linearly independent, prove that $S$ must be linearly independent.
• (c) Find (with proof) a linear function $T: \mathbb{R}^2 \to \mathbb{R}^2$ and a set $S \subset \mathbb{R}^2$ such that $T(S)$ spans $\text{Im}(T)$, but $S$ does not span $\mathbb{R}^2$.
• (d) Find (with proof) a linear function $T: \mathbb{R}^2 \to \mathbb{R}^2$ and a set $S \subset \mathbb{R}^2$ such that $S$ is linearly independent but $T(S)$ is not.

Problem 4:
Let $V$ and $W$ be vector spaces over a field $\mathbb{F}$, and let $T: V \to W$ be linear. Let $B = \{u_1, \dots, u_k\} \subset V$ be a basis for $V$ and define $T(B) = \{T(u) \mid u \in B\}$.

Which problem would you like help with, or would you like assistance with all of them step by step?

Here are some related questions you might consider:

1. What is the definition of a vector space, and how is it applied to prove that $L(U, V)$ is a vector space?
2. How do we prove that the composition of two linear maps is linear?
3. What are the conditions for a bijection and its inverse to be linear?
4. How can the concept of spanning sets and linear independence help in determining if $T_1(x) = T_2(x)$?
5. What is the relationship between a linear transformation and the image it maps to?

Tip: When proving linearity, always check for two key properties: additivity and scalar multiplication.