TBIL-LA Polynomial and Matrix Spaces (VS8)

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Section 2.8 Polynomial and Matrix Spaces (VS8)

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Learning Outcomes

Answer questions about vector spaces of polynomials or matrices.

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Subsection 2.8.1 Class Activities

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Fact 2.8.1.

Every vector space with finite dimension, that is, every vector space $V$ with a basis of the form ${{\vec{v}}_{1}, {\vec{v}}_{2}, \dots, {\vec{v}}_{n}}$ is said to be isomorphic to a Euclidean space $R^{n},$ since there exists a natural correspondance between vectors in $V$ and vectors in $R^{n} :$

c_{1} {\vec{v}}_{1} + c_{2} {\vec{v}}_{2} + \dots + c_{n} {\vec{v}}_{n} \leftrightarrow [\begin{matrix} c_{1} \\ c_{2} \\ ⋮ \\ c_{n} \end{matrix}]

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Observation 2.8.2.

We've already been taking advantage of the previous fact by converting polynomials and matrices into Euclidean vectors. Since $P_{3}$ and $M_{2, 2}$ are both four-dimensional:

4 x^{3} + 0 x^{2} - 1 x + 5 \leftrightarrow [\begin{matrix} 4 \\ 0 \\ - 1 \\ 5 \end{matrix}] \leftrightarrow [\begin{array}{cc} 4 & 0 \\ - 1 & 5 \end{array}]

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Activity 2.8.3.

Suppose $W$ is a subspace of $P_{8},$ and you know that the set ${x^{3} + x, x^{2} + 1, x^{4} - x}$ is a linearly independent subset of $W .$ What can you conclude about $W ?$

The dimension of $W$ is 3 or less.
The dimension of $W$ is exactly 3.
The dimension of $W$ is 3 or more.

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Activity 2.8.4.

Suppose $W$ is a subspace of $P_{8},$ and you know that $W$ is spanned by the six vectors

{x^{4} - x, x^{3} + x, x^{3} + x + 1, x^{4} + 2 x, x^{3}, 2 x + 1} .

What can you conclude about $W ?$

The dimension of $W$ is 6 or less.
The dimension of $W$ is exactly 6.
The dimension of $W$ is 6 or more.

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Observation 2.8.5.

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.

Since $P$ and other infinite-dimensional spaces cannot be treated as an isomorphic finite-dimensional Euclidean space $R^{n},$ vectors in such 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 2.8.2 Videos

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Figure 21. Video: Polynomial and matrix calculations

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Exploration 2.8.6.

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.

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

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Exploration 2.8.7.

Calculate a basis for the row space and a basis for the column space of the matrix

[\begin{array}{cccc} 2 & 0 & 3 & 4 \\ 0 & 1 & 1 & - 1 \\ 3 & 1 & 0 & 2 \\ 10 & - 4 & - 1 & - 1 \end{array}] .

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Exploration 2.8.8.

If you are given the values of

a, b,

and

c,

what value of

d

will cause the matrix

[\begin{array}{cc} a & b \\ c & d \end{array}]

to have rank 1?

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Subsection 2.8.6 Sample Problem and Solution

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Sample problem Example B.1.12.

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