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Discovery guide 44.1 Discovery guide

Discovery 44.1.

Let \(\funcdef{T}{\poly_3(\R)}{\matrixring_2(\R)}\) and \(\funcdef{S}{\matrixring_2(\R)}{\R^1}\) be defined by

\begin{align*} T(a x^3 + b x^2 + c x + d) \amp = \begin{bmatrix} a + b \amp b + c \\ c + d \amp d - a \end{bmatrix} \text{,} \amp S\left(\begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix}\right) \amp = a + d\text{.} \end{align*}
(a)

Are \(T\) and \(S\) linear?

(b)

What are the domain and the codomain of the composite transformation \(S \circ T\text{?}\)

(c)

Compute a general input-output formula for \(S \circ T\) similar to the formulas for \(S\) and \(T\) defined above.

Discovery 44.2.

Verify that if \(\funcdef{T}{U}{V}\) and \(\funcdef{S}{V}{W}\) are linear, then \(\funcdef{S \circ T}{U}{W}\) is also linear.

Discovery 44.3.

Consider matrix transformations \(\funcdef{T_A}{\R^3}{\R^2}\) and \(\funcdef{S_B}{\R^2}{\R^4}\) corresponding to matrices

\begin{align*} A \amp = \left[\begin{array}{rcr} 1 \amp 2 \amp -1 \\ -1 \amp 3 \amp 1 \end{array}\right] \text{,} \amp B \amp = \left[\begin{array}{rcr} 1 \amp 1 \\ 2 \amp -2 \\ -3 \amp 4 \\ 0 \amp 5 \end{array}\right]\text{.} \end{align*}
(a)

How should an image vector \((S_B \circ T_A)(\uvec{x})\) be computed?

(b)

Corollary 42.5.4 says that every transformation \(\R^3 \to \R^4\) is a matrix transformation. Based on the pattern in Task a, what is the matrix of the composite transformation \(S_B \circ T_A\text{?}\)

(c)

Describe the pattern in words: The standard matrix of a composition of matrix transformations is .

Describe the pattern in symbols: For \(\funcdef{T}{\R^n}{\R^m}\) and \(\funcdef{S}{\R^m}{\R^\ell}\text{,}\) \(\stdmatrixOf{S \circ T} = \underline{\hspace{4.545454545454546em}} \text{.}\)

Notation.

In analogy with the pattern of Discovery 44.3.c, we will use multiplication notation \(S T\) in place of composite function notation \(S \circ T\) for all linear transformations.

Discovery 44.4.

Let \(\uppermatring_2(\R)\) represent the space of \(2 \times 2\) upper triangular matrices, and let

\begin{align*} \amp \funcdef{T}{\poly_2(\R)}{\matrixring_2(\R)} \text{,} \amp \amp \funcdef{S}{\uppermatring_2(\R)}{\poly_2(\R)} \end{align*}

be defined by

\begin{gather*} T(a x^2 + b x + c) = \begin{bmatrix} a + b \amp b + c \\ 0 \amp c \end{bmatrix} \text{,}\\ S\left(\begin{bmatrix} a \amp b \\ 0 \amp c \end{bmatrix}\right) = (a - b + c) x^2 + (b - c) x + c\text{.} \end{gather*}
(a)

Compute an input-output formula similar to those above for the composite transformation \(S T\text{.}\)

(b)

Based on your result in Task a, what should we call \(S\) relative to \(T\text{?}\)

(c)

Does it work the other way? Compute an input-output formula for the composite transformation \(T S\text{.}\)

(d)

Would Task c have worked if \(S\) used the same output formula, but was defined as a transformation \(\funcdef{S}{\matrixring_2(\R)}{\poly_2(\R)}\text{?}\) Say, as

\begin{equation*} S\left(\begin{bmatrix} a \amp b \\ z \amp c \end{bmatrix}\right) = (a - b + c) x^2 + (b - c) x + c\text{?} \end{equation*}
(e)

Describe the pattern: If \(T\) is an invertible linear transformation, the domain of \(\inv{T}\) should be .

Discovery 44.5.

Consider \(\funcdef{T}{\poly_2(\R)}{\matrixring_2(\R)}\) defined by

\begin{equation*} T(a x^2 + b x + c) = \begin{bmatrix} 0 \amp a + b + c \\ 0 \amp 0 \end{bmatrix} \text{.} \end{equation*}
(a)

Based on the pattern from Discovery 44.4.e, if \(T\) is invertible, what should the domain of \(\inv{T}\) be?

(b)

Is \(T\) invertible? To decide, try to define an input-output formula for \(\funcdef{\inv{T}}{D}{\poly_2(\R)}\) (where \(D\) is the domain for \(\inv{T}\) that you identified in Task a) so that your formula “reverses” \(T\text{,}\) just as \(S\) reversed the transformation \(T\) in Discovery 44.4.

(c)

Compute the image vectors \(T(x^2)\) and \(T(1)\text{.}\)

What do the results say about the potential invertibility of \(T\text{?}\)

(d)

Compute the image vector \(T(x^2 - 1) \) (and compare with Task c).

What does the result say about the potential invertibility of \(T\text{?}\)

(e)

Describe the pattern: If a transformation \(\funcdef{T}{V}{W}\) is invertible, then the kernel of \(T\) must be .

So \(\nullity T\) must be and then \(\rank T\) must be .

Hint

For \(\rank T\text{,}\) consider the Dimension Theorem.

Discovery 44.6.
(a)

Based on Discovery 44.5.e, could a transformation \(\funcdef{T}{\matrixring_5(\R)}{\R^5}\) be invertible?

(b)

Describe the pattern: If a transformation \(\funcdef{T}{V}{W}\) is invertible, then \(\dim V\) and \(\dim W\) must satisfy: .

Discovery 44.7.

Consider matrix transformation \(\funcdef{T_A}{\R^3}{\R^3}\) corresponding to matrix

\begin{equation*} A = \left[\begin{array}{ccr} 1 \amp 2 \amp -1 \\ 0 \amp 3 \amp 1 \\ 0 \amp 1 \amp 2 \end{array}\right]\text{.} \end{equation*}
(a)

Confirm that \(T_A\) satisfies the conditions you've identified in both Discovery 44.5.e and Discovery 44.6.b.

Assuming \(T_A\) is invertible, what must the domain of \(\inv{T}_A\) be?

(b)

Corollary 42.5.4 says that every transformation \(\R^3 \to \R^3\) is a matrix transformation. Based on the calculation patterns from Discovery 44.4, we should have \((\inv{T}_A T_A)(\uvec{x}) = \uvec{x}\) for every \(\uvec{x}\) in \(\R^3\text{.}\)

So what matrix should correspond to \(\inv{T}_A\text{?}\)

(c)

Describe the pattern in words: The standard matrix of the inverse of a (square) matrix transformation is .

Describe the pattern in symbols: For invertible \(\funcdef{T}{\R^n}{\R^n}\text{,}\) \(\stdmatrixOf{\inv{T}} = \underline{\hspace{2.272727272727273em}}\text{.}\)

An isomorphism is an invertible linear transformation whose image is the whole codomain space.

Discovery 44.8.

In each of the following, provide a specific example of an isomorphism \(\funcdef{T}{V}{\R^n}\) for a specific value of \(n\text{.}\) In each case, are there multiple values of \(n\) for which this is achievable?

(a)

\(V = \matrixring_2(\R)\text{.}\)

(b)

\(V = \uppermatring_2(\R)\text{.}\)

(c)

\(V = \poly_3(\R)\text{.}\)

(d)

\(V = \poly_2(\R)\text{.}\)

(e)

\(V = \Span \{ \sin x, \cos x, e^x \}\) (as a subspace of \(F(\R)\)).

It's likely that all of your examples followed the same simple pattern of turning the inputs into outputs — what previous course concept were you using (even if unknowingly)?