Linear Assignment 6

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 $h \in \mathcal{L}(V, W)$ is both one-to-one and onto (i.e. that $h$ is an isomorphism between vector spaces).  Prove that the inverse function $h^{-1}:W\to V$ is a linear map.

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

$h^{-1}(\vec{w}) = \vec{v} \, \iff \, \vec{w} = h(\vec{v})$.

Problem 2.  (a) Let $\text{id}_{\mathbb{R}^n}$ denote the identity map $\text{id}_{\mathbb{R}^n}:\mathbb{R}^n \to \mathbb{R}^n$, which was proven to be a homomorphism in your previous assignment.  Compute the matrix representation of this map with respect to the standard bases $\mathcal{E}_n$ (for both the domain and co-domain space).  That is, compute

$\text{Rep}_{\mathcal{E}_n, \mathcal{E}_n} \, \left(\text{id}_{\mathbb{R}^n}\right) =$

The matrix you obtain is called the $n\times n$ identity matrix, and is one you’ve likely seen in previous classes.  It is often denoted by $I_n$.

(b) Compute a different matrix representation of the same identity map for $\mathbb{R}^2$, only this time use the basis $\mathcal{B} = \{\,(1,1)^T, (1,0)^T \}$ and the basis $\mathcal{D} = \{\,(0,2)^T, (1, 5)^T\,\}$.

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

In Example 2.3, we find the $2\times 2$ matrix

$H = \left(\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}\right)$.

As part of this example, the book notes that two functions $h_1$ and $h_2$ can be represented by the single matrix $H$ via the following equations:

$\text{Rep}_{\mathcal{B}_1,\mathcal{D}_1} \left(h_1\right) = H$

$\text{Rep}_{\mathcal{B}_2,\mathcal{D}_2}\left(h_2\right) = H$.

(a) Explain in your own words why these two functions are not equal, i.e. that $h_1 \neq h_2$ as functions $h_i:\mathbb{R}^2 \to \mathbb{R}^2$.

(b) Did the matrix $H$ have to represent homomorphisms in $\mathcal{L}(\mathbb{R}^2, \mathbb{R}^2)$ 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 $2\times 2$ matrix

$H = \left(\begin{array}{cc} 1 & 2 \\ 2 & 5 \end{array}\right)$

represents a non-singular homomorphism?

(c ) Is it possible that the matrix

$H = \left(\begin{array}{ccc} 1 & 2 & 3 \\ 4 & 5 & 6 \end{array}\right)$

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 $h \in \mathcal{L}(V, W)$ and that $g \in \mathcal{L}(W, U)$ where $V, W,$ and $U$ are all vector spaces.  By definition, the function $g\circ h : V \to U$.

Let $\vec{v}_1$ and $\vec{v}_2$ be arbitrary vectors in $V$.  Then

$\left(g\circ h\right)(\vec{v}_1+\vec{v}_2) = g\left(\,h(\vec{v}_1+\vec{v}_2)\,\right) = g\left(h(\vec{v}_1) + h(\vec{v}_2)\right) = g(h(\vec{v}_1)) + g(h(\vec{v}_2)) = (g\circ h)(\vec{v}_1) + (g\circ h)(\vec{v}_2)$

since both $g$ and $h$ are assumed to be linear.  Similarly, let $\vec{v} \in V$ be arbitrary and let $c \in \mathbb{R}$ be arbitrary.  Then

$\left(g\circ h\right)(c\cdot\vec{v}) = g\left(\,h(c\cdot\vec{v})\,\right) = g\left(c\cdot h(\vec{v})\,\right) = c\cdot g(h(\vec{v}) = c\cdot\,\left(g\circ h\right)(\vec{v}$

This completes the proof.  $\square$

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

$M = \left( \begin{array}{cc} 2 & 4 \\ 1 & 3 \end{array}\right)$.

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

$M^{-1} = \left( \begin{array}{cc} 3/2 & -2 \\ -1/2 & 1 \end{array}\right)$.

(b) Consider the homomorphism $h:\mathbb{R}^2\to\mathbb{R}^2$ given by

$h(x, y) = (2x + 4y, x + 3y)^T$.

Find a formula for $h^{-1}$ (assuming it exists.)

Problem 9.  Hi!