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Discovery guide 45.1 Discovery guide

First, let’s revisit Discovery 44.1.8.

Discovery 45.1.1.

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 45.1.2.

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 45.1.1.
For each, carry out the following.
  1. Choose an appropriate isomorphism from Discovery 45.1.1 and the inverse of an appropriate isomorphism from Discovery 45.1.1 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 \(W\text{.}\)
  2. Determine an input-output formula for the composite transformation \(\funcdef{T'}{\R^n}{\R^m}\) that you’ve created in the first step.
  3. 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 45.1.3.

The transformations \(T_1\) and \(T_2\) from Task (a) and Task (a) of Discovery 45.1.2 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 45.1.2 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 45.1.2? Check whether you are correct.

Discovery 45.1.5.

Once again, consider differentiation \(\funcdef{\ddx}{V}{V}\) as a linear operator on \(V = \Span \{ e^x \sin x, e^x \cos x \}\text{.}\)

(b)

Consider again your matrix for differentiation on \(V\) from Task (c) of Discovery 45.1.2. 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 antiderivative for \(f(x) = 3 e^x \sin x - e^x \cos x\text{.}\)
Let’s carry out the tasks of Discovery 45.1.1 and Discovery 45.1.2 again, but with a couple of twists.

Discovery 45.1.6.

(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 45.1.2, 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
  • the first arrow is the inverse of your transformation \(\poly_2(\R) \to \R^3\) from Task (b) (using the provided basis \(\basisfont{B}\) for \(\poly_2(\R)\)).
  • the second arrow is the identity operator; and
  • the third arrow is your transformation \(\poly_2(\R) \to \R^3\) from Task (a) (using the standard basis \(\basisfont{S}\) for \(\poly_2(\R)\));
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?
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