TBIL-LA Polynomial and Matrix Spaces (AT6)

$0 {\vec{e}}_{1} + 0 {\vec{e}}_{2} + 0 {\vec{e}}_{3} = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$
$(a + b + c) {\vec{e}}_{1} + 0 {\vec{e}}_{2} + 0 {\vec{e}}_{3} = [\begin{matrix} a + b + c \\ 0 \\ 0 \end{matrix}]$
$a {\vec{e}}_{1} + b {\vec{e}}_{2} + c {\vec{e}}_{3} = [\begin{matrix} a \\ b \\ c \end{matrix}]$

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Fact 3.6.4.

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Every vector space with finite dimension, that is, every vector space

V

with a basis of the form

{{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{n}}

has a linear bijection

T

with Euclidean space

R^{n}

that simply swaps its basis with the standard basis

{{\vec{e}}_{1}, {\vec{e}}_{2}, \dots, {\vec{e}}_{n}}

for

R^{n} :

T (c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2} + \dots + c_{n} {\vec{v}}_{n}) = c_{1} {\vec{e}}_{1} + c_{2} {\vec{e}}_{2} + \dots + c_{n} {\vec{e}}_{n} = [\begin{matrix} c_{1} \\ c_{2} \\ ⋮ \\ c_{n} \end{matrix}]

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This transformation (in fact, any linear bijection between vector spaces) is called an isomorphism, and

V

is said to be isomorphic to

R^{n} .

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Note, in particular, that every vector space of dimension

n

is isomorphic to

R^{n} .

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Activity 3.6.5.

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Consider the matrix space

M_{2, 2} = {[\begin{array}{cc} a & b \\ c & d \end{array}] | a, b, c, d \in R}

and the following set of matrices:

S = {[\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}], [\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}], [\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}], [\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}]} .

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(a)

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Does the set

S

span

M_{2, 2} ?

No; the matrix $[\begin{array}{cc} 1 & 3 \\ 2 & 4 \end{array}]$ is not a linear combination of the matrices in $S .$
No; the matrix $[\begin{array}{cc} 7 & 1 \\ 0 & - 1 \end{array}]$ is not a linear combination of the matrices in $S .$
No; the matrix $[\begin{array}{cc} - 1 & 5 \\ 2 & 9 \end{array}]$ is not a linear combination of the matrices in $S .$
Yes, every matrix in $M_{2, 2}$ is a linear combination of the matrices in $S .$

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(b)

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Is the set

S

linearly independent?

No; the matrix $[\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}] \in S$ is a linear combination of the other matrices in $S .$
No; the matrix $[\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}] \in S$ is a linear combination of the other matrices in $S .$
No; the matrix $[\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}] \in S$ is a linear combination of the other matrices in $S .$
Yes; no matrix in $S$ is a linear combination of the other matrices in $S .$

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(c)

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What statement do you think best describes the set

S = {[\begin{array}{cc} 1 & 0 \\ 0 & 0 \end{array}], [\begin{array}{cc} 0 & 1 \\ 0 & 0 \end{array}], [\begin{array}{cc} 0 & 0 \\ 1 & 0 \end{array}], [\begin{array}{cc} 0 & 0 \\ 0 & 1 \end{array}]} ?

$S$ is linearly independent
$S$ spans $M_{2, 2}$
$S$ is a basis of $M_{2, 2}$
$S$ is a basis of $R^{4}$

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(d)

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What is the dimension of

M_{2, 2} ?

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(e)

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Which Euclidean space is

M_{2, 2}

isomorphic to?

$R^{2}$
$R^{3}$
$R^{4}$
$R^{5}$

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(f)

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Describe an isomorphism

T : M_{2, 2} \to R^{?} :

T ([\begin{array}{cc} a & b \\ c & d \end{array}]) = [\begin{matrix} ? \\ ⋮ \\ ? \end{matrix}]

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Activity 3.6.6.

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Consider polynomial space

P^{4} = {a + b y + c y^{2} + d y^{3} + e y^{4} | a, b, c, d, e \in R}

and the following set:

S = {1, y, y^{2}, y^{3}, y^{4}} .

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(a)

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Does the set

S

span

P^{4} ?

No; the polynomial $1 + y^{2} + 2 y^{3}$ is not a linear combination of the polynomials in $S .$
No; the polynomial $6 + y - y^{3} + y^{4}$ is not a linear combination of the polynomials in $S .$
No; the polynomial $y^{2} + 2 y^{3} - y^{4}$ is not a linear combination of the polynomials in $S .$
Yes; every polynomial in $P^{4}$ is a linear combination of the matrices in $S .$

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(b)

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Is the set

S

linearly independent?

No; the polynomial $y^{2}$ is a linear combination of the other polynomials in $S .$
No; the polynomial $y^{3}$ is a linear combination of the other polynomials in $S .$
No; the polynomial $1$ is a linear combination of the other polynomials in $S .$
Yes; no polynomial in $S$ is a linear combination of the other polynomials in $S .$

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(c)

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What statement do you think best describes the set

S = {1, y, y^{2}, y^{3}, y^{4}} ?

$S$ is linearly independent
$S$ spans $P^{4}$
$S$ is a basis of $P^{4}$

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(d)

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What is the dimension of

P^{4} ?

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(e)

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Which Euclidean space is

P^{4}

isomorphic to?

$R^{2}$
$R^{3}$
$R^{4}$
$R^{5}$

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(f)

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Describe an isomorphism

T : P^{4} \to R^{?} :

T (a + b y + c y^{2} + d y^{3} + e y^{4}) = [\begin{matrix} ? \\ ⋮ \\ ? \end{matrix}]

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Remark 3.6.7.

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Since any finite-dimensional vector space is isomorphic to a Euclidean space

R^{n},

one approach to answering questions about such spaces is to answer the corresponding question about

R^{n} .

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Activity 3.6.8.

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Consider how to construct the polynomial

x^{3} + x^{2} + 5 x + 1

as a linear combination of polynomials from the set

{x^{3} - 2 x^{2} + x + 2, 2 x^{2} - 1, - x^{3} + 3 x^{2} + 3 x - 2, x^{3} - 6 x^{2} + 9 x + 5} .

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(a)

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Describe the vector space involved in this problem, and an isomorphic Euclidean space and relevant Eucldean vectors that can be used to solve this problem.

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(b)

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Show how to construct an appropriate Euclidean vector from an approriate set of Euclidean vectors.

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(c)

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Use this result to answer the original question.

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Observation 3.6.9.

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The space of polynomials

P

(of any degree) has the basis

{1, x, x^{2}, x^{3}, \dots},

so it is a natural example of an infinite-dimensional vector space.

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Since

P

and other infinite-dimensional vector spaces cannot be treated as an isomorphic finite-dimensional Euclidean space

R^{n},

vectors in such vector spaces cannot be studied by converting them into Euclidean vectors. Fortunately, most of the examples we will be interested in for this course will be finite-dimensional.

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Subsection 3.6.3 Individual Practice

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Activity 3.6.10.

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Let

A = [\begin{array}{ccc} - 2 & - 1 & 1 \\ 1 & 0 & 0 \\ 0 & - 4 & - 2 \\ 0 & 1 & 3 \end{array}]

and let

T : R^{3} \to R^{4}

denote the corresponding linear transformation. Note that

RREF (A) = [\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{array}] .

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The following statements are all invalid for at least one reason. Determine what makes them invalid and, suggest alternative valid statements that the author may have meant instead.

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(a)

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The matrix

A

is injective because

RREF (A)

has a pivot in each column.

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(b)

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The matrix

A

does not span

R^{4}

because

RREF (A)

has a row of zeroes.

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(c)

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The transformation

T

does not span

R^{4} .

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(d)

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The transformation

T

is linearly independent.

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Subsection 3.6.4 Videos

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Figure 39. Video: Polynomial and matrix calculations

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Exercises 3.6.5 Exercises

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Exercises available at https://tbil.org/linear-algebra/preview/exercises/#/bank/AT6/.

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Subsection 3.6.6 Mathematical Writing Explorations

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Exploration 3.6.11.

Given a matrix

M

the span of the set of all columns is the column space
the span of the set of all rows is the row space
the rank of a matrix is the dimension of the column space.

Calculate the rank of these matrices.

$[\begin{array}{ccc} 2 & 1 & 3 \\ 1 & - 1 & 2 \\ 1 & 0 & 3 \end{array}]$
$[\begin{array}{cccc} 1 & - 1 & 2 & 3 \\ 3 & - 3 & 6 & 3 \\ - 2 & 2 & 4 & 5 \end{array}]$
$[\begin{array}{ccc} 1 & 3 & 2 \\ 5 & 1 & 1 \\ 6 & 4 & 3 \end{array}]$
$[\begin{array}{ccc} 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{array}]$