Discover Linear Algebra

(a)

Suppose $A$ is a $4 \times 5$ matrix whose RREF is

Determine a basis for $\ker T_A$ as a subspace of $\R^5\text{.}$

(b)

Connect to previous concepts: $\ker T_A$ is the same as the of $A\text{.}$

(a)

$\funcdef{T}{\matrixring_n(\R)}{\matrixring_n(\R)}$ by $T(A) = A - \utrans{A}\text{.}$

(b)

Evaluation of polynomials at fixed $x$-value $x = a\text{:}$

$\funcdef{E_a}{\poly(\R)}{\R^1}$ 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 $I_{a,b}(f) = \integral{a}{b}{f(x)}{x}\text{.}$

(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

(a)

Apply the definition: An $m$-dimensional vector $\uvec{b}$ is in $\im T_A$ precisely when .

(b)

Connect to previous concepts: $\im T_A$ is the same as the of $A\text{.}$

(c)

If you remembered the relevant previous concept in Task b, then hopefully you also remember how to determine a basis for that special space attached to matrix $A\text{.}$

Suppose $A$ is a $4 \times 5$ matrix whose RREF is

Describe how to determine a basis for $\im T_A$ as a subspace of $\R^4\text{.}$

(d)

The matrix in Task c is the same as the one in Discovery 43.1.a. How does $\dim (\ker T_A)$ relate to $\dim (\im T_A)\text{?}$

(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)

Compute the images of the standard basis vectors:

Discovery 43.5.c 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 Discovery 43.3.c.

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 $\funcdef{T}{V}{W}\text{,}$ .