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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.
(d)
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 = \begin{abmatrix}{rcr}
1 \amp 2 \amp -1 \\
-1 \amp 3 \amp 1
\end{abmatrix}
\text{,}
\amp
B \amp = \begin{abmatrix}{rcr}
1 \amp 1 \\
2 \amp -2 \\
-3 \amp 4 \\
0 \amp 5
\end{abmatrix}\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
.
In symbols: For
\(\funcdef{T}{\R^n}{\R^m}\) and
\(\funcdef{S}{\R^m}{\R^\ell}\text{,}\) \(\stdmatrixOf{S \circ T} = \fillinmath{XXXXXXXXXX} \text{.}\)
Notation. In analogy with the pattern of
Task c of
Discovery 44.3 , 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
Task e of
Discovery 44.4 , 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
.
Discovery 44.6 .
(a) Based on
Task e of
Discovery 44.5 , 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 = \begin{abmatrix}{ccr}
1 \amp 2 \amp -1 \\
0 \amp 3 \amp 1 \\
0 \amp 1 \amp 2
\end{abmatrix}\text{.}
\end{equation*}
(a) 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
.
In symbols: For invertible
\(\funcdef{T}{\R^n}{\R^n}\text{,}\) \(\stdmatrixOf{\inv{T}} = \fillinmath{XXXXX}\text{.}\)
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)?
