## Background

The point of this assignment is to review and further develop many of the ideas explored in the previous assignment, namely concepts concerning the Rank-Nullity Theorem, isomorphisms between vector spaces, matrix representations of homomorphisms, etc.

This material comes primarily from Chapter III, pages 202-263. (Our upcoming exam will cover material from Chapters I, II, and III.)

## The Actual Assignment

**Problem 1**. Suppose is both one-to-one and onto (i.e. that is an isomorphism between vector spaces). Prove that the inverse function is a linear map.

Note: the definition of an inverse function is more descriptive than formula-based. One defines

.

**Problem 2**. (a) Let denote the identity map , which was proven to be a homomorphism in your previous assignment. Compute the matrix representation of this map with respect to the standard bases (for both the domain and co-domain space). That is, compute

The matrix you obtain is called *the identity matrix*, and is one you’ve likely seen in previous classes. It is often denoted by .

(b) Compute a different matrix representation of the same identity map for , only this time use the basis and the basis .

**Problem 3**. On page 213 our textbook discusses how, conversely, every matrix can be thought of as representing a homomorphism from an -dimensional space to an -dimensional space.

In Example 2.3, we find the matrix

.

As part of this example, the book notes that two functions and can be represented by the single matrix via the following equations:

.

(a) Explain in your own words why these two functions are not equal, i.e. that as functions .

(b) Did the matrix *have* to represent homomorphisms in or could it have been used to represent homomorphisms between *other* vector spaces? If yes, explain why; if not, provide an example.

**Problem 4**. Read and explain Corollary 2.6 (on page 216) in your own words.

**Problem 5**. (a) What does it mean to say that a linear map is *non-singular*?

(b) Is it possible that the matrix

represents a non-singular homomorphism?

(c ) Is it possible that the matrix

represents a non-singular homomorphism?

**Problem 6**. Read the following proof and then write down the theorem or Lemma that it demonstrates.

Theorem: ______________________________________________

proof: Suppose and that where and are all vector spaces. By definition, the function .

Let and be arbitrary vectors in . Then

since both and are assumed to be linear. Similarly, let be arbitrary and let be arbitrary. Then

This completes the proof.

**Problem 7**. Explain why matrix multiplication is defined the way it is (use words like “represent” and “composition” in your explanation).

**Problem 8**. Consider the matrix

.

(a) Confirm that the inverse of this matrix is given by

.

(b) Consider the homomorphism given by

.

Find a formula for (assuming it exists.)

**Problem 9**. Hi!