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\)
Discovery guide 45.1 Discovery guide
Discovery 45.1 .
In each of the following, determine an input-output formula for the isomorphism
\(V \to W\) that sends the standard basis for the domain space to the standard basis for the codomain space. Then determine an input-output formula for the inverse isomorphism.
(a)
\(V = \matrixring_2(\R)\text{,}\) \(W = \R^4\text{.}\)
(b)
\(V = \uppermatring_2(\R)\text{,}\) \(W = \R^3\text{.}\)
(c)
\(V = \poly_3(\R)\text{,}\) \(W = \R^4\text{.}\)
(d)
\(V = \poly_2(\R)\text{,}\) \(W = \R^3\text{.}\)
(e)
\(V = \Span \{ e^x \sin x, e^x \cos x \}\) (as a subspace of
\(F(\R)\) ),
\(W = \R^2\text{.}\)
Discovery 45.2 .
In each of the following, you are given a transformation
\(\funcdef{T}{V}{W}\text{,}\) where
\(V,W\) are spaces from various tasks in
Discovery 45.1 .
For each, carry out the following.
Choose an appropriate isomorphism from
Discovery 45.1 and the
inverse of an appropriate isomorphism from
Discovery 45.1 to chain together with
\(T\) to create a transformation
\begin{equation*}
\R^n \xrightarrow{\invcoordmapplain{\basisfont{S}_V}} V \xrightarrow{T} W \xrightarrow{\coordmapplain{\basisfont{S}_W}} \R^m \text{,}
\end{equation*}
for appropriate values of
\(n\) and
\(m\text{,}\) where
\(\basisfont{S}_V\) is the standard basis of
\(V\) and
\(\basisfont{S}_W\) is the standard basis for
\(W\text{.}\)
Determine an input-output formula for the composite transformation \(\funcdef{T'}{\R^n}{\R^m}\) that you’ve created in the first step.
Every transformation \(\R^n \to \R^m\) is a matrix transformation. Determine the standard matrix \(\stdmatrixOf{T'}\) for your transformation from the second step. (Recall that you can do this from your input output formulas, or by determining the outputs for standard basis vectors.)
(a)
\(\funcdef{T_1}{\matrixring_2(\R)}{\poly_2(\R)}\) by
\(\displaystyle T_1\left(\begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix}\right) = -d + (a + b + c) x + (a + b) x^2\text{.}\)
(b)
\(\funcdef{T_2}{\poly_2(\R)}{\uppermatring_2(\R)}\) by
\(\displaystyle T_2(a_0 + a_1 x + a_2 x^2) = \begin{bmatrix} a_2 - a_1 \amp a_0 + a_1 \\ 0 \amp a_0 - a_2 \end{bmatrix}\text{.}\)
(c) For
\(V = \Span \{ e^x \sin x, e^x \cos x \}\text{,}\) let
\(T_3 = \ddx\) be differentiation
\(V \to V\text{.}\)
Discovery 45.3 .
The transformations
\(T_1\) and
\(T_2\) from
Task a and
Task a of
Discovery 45.2 can be composed to create a transformation
\(\funcdef{T_2 T_1}{\matrixring_2(\R)}{\uppermatring_2(\R)}\text{.}\)
(a) Repeat the three steps described in the introduction to
Discovery 45.2 to create a matrix corresponding to
\(T_2 T_1\text{.}\)
(b) How do you think your matrix for the composition
\(T_2 T_1\) relates to the matrices for
\(T_1\) and
\(T_2\) that you already calculated in
Discovery 45.2 ? Check whether you are correct.
Discovery 45.4 .
Figure out how to use the pattern you discovered in
Discovery 45.3 , applied using your matrix from
Task c of
Discovery 45.2 , to compute the
second derivative of
\(f(x) = 3 e^x \sin x - e^x \cos x\text{.}\)
Discovery 45.5 .
Once again, consider differentiation
\(\funcdef{\ddx}{V}{V}\) as a linear operator on
\(V = \Span \{ e^x \sin x, e^x \cos x \}\text{.}\)
(a) What is
\(\ker \ddx\) on this domain?
What does this say about differentiation on this domain?
(b) Consider again your matrix for differentiation on
\(V\) from
Task c of
Discovery 45.2 . Do you think you could have come to the same conclusions about this operator as in
Task a from some property of the corresponding matrix?
(c) Figure out how to use your matrix for differentiation on
\(V\) to compute an
anti derivative for
\(f(x) = 3 e^x \sin x - e^x \cos x\text{.}\)
Discovery 45.6 .
(a) Remind yourself how your input-output formulas worked for
\(\poly_2(\R) \to \R^3\) and its inverse in
Task d of
Discovery 45.1 .
(b)
In the same way, determine a new input-output formula for a transformation \(\poly_2(\R) \to \R^3\) that sends the basis
\begin{equation*}
\basisfont{B} = \{ x^2 - 1, x + 1, x \}
\end{equation*}
to the standard basis for \(\R^3\text{,}\) along with an input-output formula for its inverse.
(c)
Similarly to
Discovery 45.2 , create a matrix for the chain of transformations
\begin{equation*}
\R^3 \xrightarrow{\invcoordmap{B}} \poly_2(\R) \xrightarrow{I} \poly_2(\R) \xrightarrow{\coordmap{S}} \R^3 \text{,}
\end{equation*}
where
Look at the columns of your matrix, compared to the basis vectors in \(\basisfont{B}\text{.}\) What matrix corresponding to a previous concept do you think you just calculated?
(d)
Suppose you repeated
Task c for the reverse composition:
\begin{equation*}
\R^3 \xrightarrow{\invcoordmap{S}} \poly_2(\R) \xrightarrow{I} \poly_2(\R) \xrightarrow{\coordmap{B}} \R^3 \text{.}
\end{equation*}
What matrix would you have calculated in that case?
