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Discovery guide 43.1 Discovery guide
Attached to every linear transformation \(\funcdef{T}{V}{W}\) is a pair of important subspaces, one in the domain space and one in the codomain space.
kernel of \(T\)
the collection of all vectors
\(\uvec{v}\) in the domain space
\(V\) for which
\(T(\uvec{v}) = \zerovec_W\)
\(\ker T\)
notation for the kernel of
\(T\)
image of \(T\)
the collection of all image vectors
\(T(\uvec{v})\) in the codomain space
\(W\)
\(\im T\)
notation for the image of
\(T\)
Note that a vector \(\uvec{w}\) in \(W\) is in \(\im T\) precisely when there exists at least one \(\uvec{v}\) in \(V\) with \(T(\uvec{v}) = \uvec{w}\text{.}\)
Discovery 43.1 . Kernel of a matrix transformation.
(a)
(b) Connect to previous concepts.
Discovery 43.2 . Kernel examples.
In each of the following, describe the kernel vectors in words.
Try to use a more meaningful description than just “the vectors that evaluate to zero in the transformation”.
(a)
\(\funcdef{T}{\matrixring_n(\R)}{\matrixring_n(\R)}\) by
\(T(A) = A - \utrans{A}\text{.}\)
(b) Polynomial evaluation.
\(\funcdef{E_a}{\poly(\R)}{\R^1}\) by
\(E_a(p) = p(a) \text{.}\)
(c) Differentiation.
Let
\(F(a,b)\) represent the space of functions defined on the interval
\(a \lt x \lt b\text{,}\) and let
\(D(a,b)\) represent the subspace of
\(F(a,b)\) consisting of
differentiable functions.
Consider
\(\funcdef{\ddx}{D(a,b)}{F(a,b)}\) by
\(\ddx(f) = f'\text{.}\)
(d) Integration.
Let
\(C[a,b]\) represent the space of
continuous functions defined on the interval
\(a \le x \le b\text{.}\)
Consider
\(\funcdef{I_{a,b}}{C[a,b]}{\R^1}\) by
\(I_{a,b}(f) = \integral{a}{b}{f(x)}{x}\text{.}\)
Discovery 43.3 . Determining kernel basis.
For each of the provided transformations of \(\matrixring_{2}(\R)\text{,}\) determine a basis for \(\ker T\) by carrying out the following steps.
Starting with an arbitrary matrix
\begin{equation*}
X = \begin{bmatrix} a \amp b \\ c \amp d \end{bmatrix}
\end{equation*}
in
\(\matrixring_{2}(\R)\text{,}\) determine conditions on parameters
\(a,b,c,d\) so that
\(T(X) = \zerovec \text{.}\)
Use those conditions to reduce the number of parameters required to describe an arbitrary matrix in \(\ker T\text{.}\)
Determine the basis vector associated to each in the reduced collection of parameters.
Aside: Compare.
(a)
\(\funcdef{T = \trace}{\matrixring_{2}(\R)}{\R^1}\text{.}\)
(b)
\(\funcdef{T}{\matrixring_{2}(\R)}{\matrixring_{2}(\R)}\) by
\(T(A) = A - \utrans{A}\text{.}\)
(c)
\(\funcdef{T = L_B}{\matrixring_{2}(\R)}{\matrixring_{2}(\R)}\) by \(L_B(A) = B A\text{,}\) where
\begin{equation*}
B = \begin{bmatrix} 1 \amp 1 \\ 0 \amp 0 \end{bmatrix} \text{.}
\end{equation*}
Discovery 43.4 . Image of a matrix transformation.
(a) Apply the definition.
(b) Connect to previous concepts.
(c) (d)
Discovery 43.5 . Describing images.
Once again, for transformation
\(\funcdef{T}{V}{W}\text{,}\) a vector
\(\uvec{w}\) in
\(W\) is in
\(\im T\) precisely when there exists a vector
\(\uvec{v}\) in
\(V\) so that
\(T(\uvec{v}) = \uvec{w}\text{.}\)
Suppose the domain space
\(V\) is finite-dimensional with spanning set
\(S = \{\uvec{v}_1,\uvec{v}_2,\uvec{v}_3\}\text{.}\)
(a) Reword the above definition of
\(\im T\) using these spanning vectors:
a vector
\(\uvec{w}\) in
\(W\) is in
\(\im T\) precisely when there exists a
of
\(\uvec{v}_1,\uvec{v}_2,\uvec{v}_3\) so that
.
(b) Reword the definition again in terms of the spanning
image vectors:
a vector
\(\uvec{w}\) in
\(W\) is in
\(\im T\) precisely when there exists a
of
\(T(\uvec{v}_1),T(\uvec{v}_2),T(\uvec{v}_3)\) so that
.
(c) Summarize.
If
\(\{\uvec{v}_1,\uvec{v}_2,\dotsc,\uvec{v}_m\}\) is a spanning set for
\(V\text{,}\) then
\(\{T(\uvec{v}_1),T(\uvec{v}_2),\dotsc,T(\uvec{v}_m)\}\) is
for
\(\im T\text{.}\)
A spanning set is fine, but a basis is better.
Discovery 43.6 . Image basis example.
Consider again the transformation
\(\funcdef{T = L_B}{\matrixring_{2}(\R)}{\matrixring_{2}(\R)}\) from
Task c of
Discovery 43.3 , defined by
\(L_B(A) = B A\) for
\begin{equation*}
B = \begin{bmatrix} 1 \amp 1 \\ 0 \amp 0 \end{bmatrix} \text{.}
\end{equation*}
Also recall the standard basis for \(\matrixring_{2}(\R)\text{:}\)
\begin{align*}
E_{11} \amp = \begin{bmatrix} 1 \amp 0 \\ 0 \amp 0 \end{bmatrix}, \amp
E_{12} \amp = \begin{bmatrix} 0 \amp 1 \\ 0 \amp 0 \end{bmatrix}, \amp
E_{21} \amp = \begin{bmatrix} 0 \amp 0 \\ 1 \amp 0 \end{bmatrix}, \amp
E_{22} \amp = \begin{bmatrix} 0 \amp 0 \\ 0 \amp 1 \end{bmatrix}\text{.}
\end{align*}
(a)
Compute the images of the standard basis vectors:
\begin{equation*}
T(E_{11}), \;\; T(E_{12}), \;\; T(E_{21}), \;\; T(E_{22}) \text{.}
\end{equation*}
Task c of
Discovery 43.5 says these image vectors should form a spanning set for
\(\im L_B\text{.}\) Do they form a basis for
\(\im L_B\text{?}\)
(b) Replace the first two standard basis vectors for
\(V = \matrixring_{2}(\R)\) with your two basis vectors for
\(\ker L_B\) that you computed in
Task c of
Discovery 43.3 .
Write
\(A_1,A_2\) for these kernel vectors. Is
\(\{A_1,A_2,E_{21},E_{22}\}\) still a basis for the domain space
\(V = \matrixring_{2}(\R)\text{?}\)
If so, then
\(\{T(A_1),T(A_2),T(E_{21}),T(E_{22})\}\) should again be a spanning set for
\(\im L_B\text{.}\) Is it a basis for
\(\im L_B\text{?}\) If not, can it easily be
reduced to a basis for
\(\im L_B\text{?}\)
(c) Summarize the pattern.
To determine a basis for the image of transformation
\(\funcdef{T}{V}{W}\text{,}\) .
Discovery 43.7 .
Create a linear transformation
\(\funcdef{T}{\poly_3(\R)}{\matrixring_2(\R)}\) that has kernel
precisely \(\Span \{ 1 + x^2, 1 - x^3 \}\text{.}\)
Hint .
A transformation does not have to be specified by a formula; see
Procedure 42.3.1 .
