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 \(\IR^n\text{:}\)
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:
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.
Activity2.8.4.
Suppose \(W\) is a subspace of \(\P_8\text{,}\) and you know that \(W\) is spanned by the six vectors
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.
Subsection2.8.1Videos
Figure2.8.6.Video: Polynomial and matrix calculations