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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:

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**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.

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**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 \( T_1(x) = T_2(x) \).

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**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.

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**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 \( T(B) = \{T(u) \mid u \in B\} \).

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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.

The original problem set related to linear algebra and vector spaces as shown in the image.

This problem set explores advanced linear algebra concepts such as vector spaces, linear maps, and their properties. Topics include proving the linearity of compositions, understanding the relationship between spanning sets and linear independence, and analyzing linear bijections and their inverses.

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