Advanced Linear Algebra 10: Linear Forms

Recorded Friday, February 4.

A second course in linear algebra covering vector spaces and matrix decompositions taught by Dr. Anthony Bosman.

Full Course:

The lectures closely follow 'Advanced Linear and Matrix Algebra' by Johnston:

Subscribe:

Learn more about the Andrews University math department:

[rocksbox]

For anyone confused by dual space, I'm understanding it as the space of all ways of assigning a "measure" to each part of the basis of the original vector space. For example, if we have the vector space of polynomials on P^3, with basis vectors {1, x, x^2, x^3}, then one way of assigning measure is with the integral from 0 to 1. This produces [1, 1/2, 1/3, 1/4] when applied to the standard basis of P^3 mentioned earlier. Another way to assign a measure is by just plugging in the number 2, so [1, 2, 4, 8]. Another measure is to just 0 out every basis vector of the original space, [0 0 0 0] Observe that this corresponds to each member of the original basis. Also observe that these measures are all linear transformations, i.e. if you sum two vectors and pass it into these measures, they will return the measure of each vector summed together (follows from the need of these measures to be represented by dot product). Now, what's interesting is that the dual is also in it's own right a vector space. We can add two different methods of measure together and multiply a method of measure by a scalar to get another method of measure. This implies then that we can also take the dual of this, i.e. a measure of measures lol.

[studentbg]

This is so cool. Such a nice and cool concept. This video was phenomenal too. I read about linear forms in my linear algebra textbook and I didn't understand what was going on, then I watched this video and I understood everything. Not only that I think the concept is cool and I saw that is is isomorphic before he said it. I like this professor and how he explains it. Studying university is easier in today's day and age because you can watch lessons from other professors from all over the world explaining the same topics.

[thewickedone]

One of the best videos I watched on the topic, really made the concept click for me finally

[godfreypigott]

Is there an algebraic way to define subspaces of the dual space, other than just the span of basis vectors, and in what applications would these subspaces arise?

[chilling]

I think it could be confusing calling the action of a linear functional w on a vector v "w dot v": it can give the false impression that v and w live in the same space.

[ktayeb]

This is Lax_Milgran theorem in finit dimension