DLA Discovery guide

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Discovery guide 42.1 Discovery guide

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Discovery 42.1.

An $m \times n$ real matrix $A$ creates a function $\funcdef{T_A}{\R^n}{\R^m}$ by matrix multiplication:

$\begin{equation*} T_A(\uvec{x}) = A \uvec{x} \text{.} \end{equation*}$

We will call such a function a matrix transformation $\R^n \to \R^m\text{.}$

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(a)

Write out linear input-output component formulas for function $T_A$ associated to matrix

$\begin{equation*} A = \left[\begin{array}{rrr} 1 \amp 2 \amp -3 \\ 2 \amp -1 \amp 5 \end{array}\right] \text{,} \end{equation*}$

so that $\uvec{w} = T_A(\uvec{x})\text{.}$

$\begin{equation*} \left\{\begin{array}{rcrcrcr} w_1 \amp = \amp \underline{\hspace{0.909090909090909em}} x_1 \amp + \amp \underline{\hspace{0.909090909090909em}} x_2 \amp + \amp \underline{\hspace{0.909090909090909em}} x_3 \text{,} \\ w_2 \amp = \amp \underline{\hspace{0.909090909090909em}} x_1 \amp + \amp \underline{\hspace{0.909090909090909em}} x_2 \amp + \amp \underline{\hspace{0.909090909090909em}} x_3 \text{.} \end{array}\right. \end{equation*}$

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(b)

Determine the matrix $B$ so that the linear input-ouput component formulas below correspond to a matrix transformation $\uvec{w} = T_B(\uvec{x})\text{.}$

$\begin{equation*} \left\{\begin{array}{rcrcr} w_1 \amp = \amp 3 x_1 \amp - \amp x_2 \\ w_2 \amp = \amp 5 x_1 \amp + \amp 5 x_2 \\ w_3 \amp = \amp \amp + \amp 7 x_2 \\ w_4 \amp = \amp - x_1 \amp + \amp x_2 \end{array}\right. \end{equation*}$

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(c)

Suppose you know that matrix transformation $\funcdef{T_C}{\R^3}{\R^3}$ satisfies

$\begin{align*} T_C(\uvec{e}_1) \amp = \left[\begin{array}{r} 2 \\ -3 \\ 5 \end{array}\right] \text{,} \amp T_C(\uvec{e}_2) \amp = \left[\begin{array}{r} -7 \\ 11 \\ 13 \end{array}\right] \text{,} \amp T_C(\uvec{e}_3) \amp = \left[\begin{array}{r} 17 \\ 19 \\ -23 \end{array}\right] \text{.} \end{align*}$

Do you have enough information to determine matrix $C\text{?}$

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A function between two “spaces” of the same kind is often referred to as a morphism. Just as we used

$\R^n$ as the model for the ten vector space axioms, and used the dot products on

$\R^n$ and

$\C^n$ as the models for the four inner product space axioms, we will use matrix transformations

$\R^n \to \R^m$ as the model for the desired properties of vector space morphisms.

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Discovery 42.2.

Suppose $\funcdef{T_A}{\R^n}{\R^m}$ is the matrix transformation associated to $m \times n$ matrix $A\text{,}$ so that

$\begin{equation*} T_A(\uvec{x}) = A \uvec{x} \text{.} \end{equation*}$

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(a)

How does $T_A$ interact with the vector operations of the domain space $\R^n$ and the codomain space $\R^m\text{?}$

That is, how does $T_A$ interact with

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(i)

vector addition?

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(ii)

scalar multiplication?

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(iii)

linear combinations?

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(iv)

negatives?

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(v)

the zero vector?

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(b)

Which of the patterns from Task a can be deduced from others of the patterns?

Based on this, which of these patterns should be designated as the basic axioms of vector space morphisms?

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A function

$\funcdef{T}{V}{W}$ between abstract vector spaces

$V,W$ that satisfies the axioms we have identified in Discovery 42.2.b will be called a linear transformation (or a vector space homomorphism).

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Discovery 42.3.

In each of the following, determine whether the provided vector space function is a linear transformation.

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(a)

Left-multiplication by $m \times n$ matrix $A\text{:}$

$\funcdef{L_A}{\matrixring_{n \times \ell}(\R)}{\matrixring_{m \times \ell}(\R)}$ by $L_A(X) = A X\text{.}$

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(b)

Right-multiplication by $m \times n$ matrix $A\text{:}$

$\funcdef{R_A}{\matrixring_{\ell \times m}(\R)}{\matrixring_{\ell \times n}(\R)}$ by $R_A(X) = X A\text{.}$

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(c)

Translation by a fixed nonzero vector $\uvec{a}$ in vector space $V\text{:}$

$\funcdef{t_{\uvec{a}}}{V}{V}$ by $t_{\uvec{a}}(\uvec{v}) = \uvec{v} + \uvec{a} \text{.}$

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(d)

Multiplication by a fixed scalar $a$ in vector space $V\text{:}$

$\funcdef{m_a}{V}{V}$ by $m_a(\uvec{v}) = a \uvec{v} \text{.}$

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(e)

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

$\funcdef{E_a}{\poly(\R)}{\R^1}$ by $E_a(p) = p(a) \text{.}$

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(f)

Determinant of square matrices: $\funcdef{\det}{\matrixring_n(\R)}{\R^1}\text{.}$

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(g)

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 $\ddx(f) = f'\text{.}$

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(h)

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 $I_{a,b}(f) = \integral{a}{b}{f(x)}{x}\text{.}$

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Discovery 42.4.

Suppose $V$ is a finite-dimensional vector space with

$\begin{equation*} V = \Span \{ \uvec{v}_1, \uvec{v}_2, \uvec{v}_3 \} \text{,} \end{equation*}$

and $\funcdef{T}{V}{\R^2}$ is a linear transformation such that

$\begin{align*} T(\uvec{v}_1) \amp = (1,2) \text{,} \amp T(\uvec{v}_2) \amp = (3,-5) \text{,} \amp T(\uvec{v}_3) \amp = (0,4)\text{.} \end{align*}$

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(a)

Based on this information, can you determine $T(3 \uvec{v}_1 - \uvec{v}_2 + 5 \uvec{v}_3)\text{?}$

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(b)

Would you be able to answer Task a for other linear combinations of $\uvec{v}_1,\uvec{v}_2,\uvec{v}_3\text{?}$

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(c)

Describe the pattern: in order to be able to compute every output of a linear transformation, the only output information required is .

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Discovery 42.5.

Suppose $\funcdef{T}{\R^3}{\R^3}$ is a linear transformation such that

$\begin{align*} T(\uvec{e}_1) \amp = \left[\begin{array}{r} 2 \\ -3 \\ 5 \end{array}\right] \text{,} \amp T(\uvec{e}_2) \amp = \left[\begin{array}{r} -7 \\ 11 \\ 13 \end{array}\right] \text{,} \amp T(\uvec{e}_3) \amp = \left[\begin{array}{r} 17 \\ 19 \\ -23 \end{array}\right] \text{.} \end{align*}$

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(a)

Do you know any other linear transformation $\R^3 \to \R^3$ that has the same outputs for the standard basis vectors as inputs?

Hint

Look back at Discovery 42.1.c.

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(b)

Based on Task a, and in light of Discovery 42.4.c, what can you say about $T\text{?}$

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(c)

Describe the pattern: every linear transformation $\R^n \to \R^m$ is effectively .

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Discovery 42.6.

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(a)

Suppose $\funcdef{T}{\R^n}{\R^1}$ is a linear transformation. What size of matrix would represent this linear transformation?

What word would we normally use to describe a matrix of those dimensions, instead of “matrix”?

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(b)

Describe the pattern: every linear transformation $\R^n \to \R^1$ corresponds to a .

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Discovery 42.7.

For vector spaces $V,W\text{,}$ let $L(V,W)$ represent the collection of all linear transformations $V \to W\text{.}$

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(a)

How could transformations in $L(V,W)$ be added?

That is, if $\funcdef{T_1,T_2}{V}{W}$ are objects in $L(V,W)\text{,}$ what transformation should $T_1 + T_2$ represent?

$\begin{equation*} (T_1 + T_2)(\uvec{v}) = \underline{\hspace{9.090909090909092em}} \end{equation*}$

Is the sum transformation $T_1 + T_2$ still in $L(V,W)\text{?}$ (i.e. Is it still linear?)

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(b)

How could transformations in $L(V,W)$ be scalar multiplied?

That is, if $\funcdef{T}{V}{W}$ is an object in $L(V,W)\text{,}$ what transformation should $k T$ represent for scalar $k\text{?}$

$\begin{equation*} (k T)(\uvec{v}) = \underline{\hspace{9.090909090909092em}} \end{equation*}$

Is the scaled transformation $T$ still in $L(V,W)\text{?}$ (i.e. Is it still linear?)

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(c)

Is $L(V,W)$ a vector space under the operations of addition and scalar multiplication of linear transformations?

That is, do your operations satisfy the ten Vector space axioms?