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!

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