Math Problem Statement

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

Solution

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

  • (a) Show that L(U,V)L(U, V) is a vector space. (Optional, ungraded)
  • (b) Suppose T1:UVT_1: U \to V and T2:VWT_2: V \to W are linear. Prove that T2T1T_2 \circ T_1 is linear.
  • (c) Suppose f:UVf: U \to V is a linear bijection and let f1:VUf^{-1}: V \to U be its inverse. Prove that f1f^{-1} is linear.

Problem 2:
Let UU and VV be vector spaces over a field F\mathbb{F}, and let T1:UVT_1: U \to V and T2:UVT_2: U \to V be linear. Suppose S={u1,,uk}US = \{u_1, \dots, u_k\} \subset U and that for all sSs \in S we have T1(s)=T2(s)T_1(s) = T_2(s). Prove that for all xspan(S)x \in \text{span}(S) we have that T1(x)=T2(x)T_1(x) = T_2(x).


Problem 3:
Let VV and WW be vector spaces over a field F\mathbb{F}, and let T:VWT: V \to W be linear. Let S={u1,,uk}VS = \{u_1, \dots, u_k\} \subset V and define T(S)={T(u)uS}T(S) = \{T(u) \mid u \in S\}.

  • (a) If SS spans VV, prove that T(S)T(S) spans the image of TT.
  • (b) If T(S)T(S) is linearly independent, prove that SS must be linearly independent.
  • (c) Find (with proof) a linear function T:R2R2T: \mathbb{R}^2 \to \mathbb{R}^2 and a set SR2S \subset \mathbb{R}^2 such that T(S)T(S) spans Im(T)\text{Im}(T), but SS does not span R2\mathbb{R}^2.
  • (d) Find (with proof) a linear function T:R2R2T: \mathbb{R}^2 \to \mathbb{R}^2 and a set SR2S \subset \mathbb{R}^2 such that SS is linearly independent but T(S)T(S) is not.

Problem 4:
Let VV and WW be vector spaces over a field F\mathbb{F}, and let T:VWT: V \to W be linear. Let B={u1,,uk}VB = \{u_1, \dots, u_k\} \subset V be a basis for VV and define T(B)={T(u)uB}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)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 T1(x)=T2(x)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.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Vector Spaces
Linear Transformations
Basis
Spanning Sets
Linear Independence
Inverse Functions

Formulas

Composition of Linear Maps
Span and Basis of Vector Spaces
Properties of Linear Transformations

Theorems

Properties of Linear Maps
Bijective Mappings and Inverses
Spanning and Basis Theorem
Linear Independence Theorem

Suitable Grade Level

Undergraduate (Linear Algebra Course)