Discover Linear Algebra

(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.

(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 $\funcdef{S}{\R^m}{\R^\ell}\text{,}$ $\stdmatrixOf{S \circ T} = \underline{\hspace{4.545454545454546em}} \text{.}$

(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

(e)

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

(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 .

(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: .

(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 $\funcdef{T}{\R^n}{\R^n}\text{,}$ $\stdmatrixOf{\inv{T}} = \underline{\hspace{2.272727272727273em}}\text{.}$

(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 $F(\R)$).