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Linear Algebra for Team-Based Inquiry Learning
2022 Edition
| Steven Clontz |
Drew Lewis |
| University of South Alabama |
University of South Alabama |
|
|
August 2, 2022
Section 2.8: Polynomial and Matrix Spaces (VS8)
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 \(\IR^n\text{,}\) since there exists a natural correspondance between vectors in \(V\) and vectors in \(\IR^n\text{:}\)
\begin{equation*}
c_1\vec v_1+c_2\vec v_2+\dots+c_n\vec v_n
\leftrightarrow
\left[\begin{array}{c}
c_1\\c_2\\\vdots\\c_n
\end{array}\right]
\end{equation*}
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:
\begin{equation*}
4x^3+0x^2-1x+5
\leftrightarrow
\left[\begin{array}{c}
4\\0\\-1\\5
\end{array}\right]
\leftrightarrow
\left[\begin{array}{cc}
4&0\\-1&5
\end{array}\right]
\end{equation*}
Activity 2.8.3 (~5 min)
Suppose \(W\) is a subspace of \(\P_8\text{,}\) and you know that the set \(\{ x^3+x, x^2+1, x^4-x \}\) is a linearly independent subset of \(W\text{.}\) What can you conclude about \(W\text{?}\)
The dimension of \(W\) is 3 or less.
The dimension of \(W\) is exactly 3.
The dimension of \(W\) is 3 or more.
Activity 2.8.4 (~5 min)
Suppose \(W\) is a subspace of \(\P_8\text{,}\) and you know that \(W\) is spanned by the six vectors
\begin{equation*}
\{ x^4-x,x^3+x,x^3+x+1,x^4+2x,x^3,2x+1\}.
\end{equation*}
What can you conclude about
\(W\text{?}\)
The dimension of \(W\) is 6 or less.
The dimension of \(W\) is exactly 6.
The dimension of \(W\) is 6 or more.
Observation 2.8.5
The space of polynomials \(\P\) (of any degree) has the basis \(\{1,x,x^2,x^3,\dots\}\text{,}\) 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 \(\IR^n\text{,}\) 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.