Advanced Linear Algebra, Lecture 1.5: Dual vector spaces

Advanced Linear Algebra, Lecture 1.5: Dual vector spaces

The dual of a vector space X over K is the space X' of all linear scalar functions from X to K, which are also called co-vectors or dual vectors. When dim(X)=n is finite, then X and X' are isomorphic. We can think about vectors as length-n column vectors, and dual vectors as length-n row vectors. The function l(x) is simply the scalar product of these, so we can denote it as (l,x)=l(x). We conclude with an example of an infinite-dimensional vector space that has a dual vector that is not of this form.

[tuatarian]

Hi, I wanted to thank you for this video. I've been working through Lax's book and was not able to understand how the bilinear scalar product notation for a (linear) function call provided a "natural isomorphism" between X'' and X, but your explanation about choosing which of the (l, x) to keep fixed cleared up my misunderstandings perfectly. Thank you!

[annemilward]

Really enjoying the lecture series, many thanks. Can I ask a question? At 5:17 second example: the sum (i=1) in the derivatives starts at one but in your handwritten green example you add in f(0). Should the sum be i=0 or f(0) not included. I am worrying I am missing something deeper!