Advanced Linear Algebra, Lecture 3.7: Tensors

Advanced Linear Algebra, Lecture 3.7: Tensors

The easiest way to motivate the tensor product of U and V is to think of U as a vector space of polynomials in x, and V as a vector space of polynomials in y. Then U⊗V is the vector space spanned by x^iy^j, of dimension (dim U)(dim V). We formalized the "product" of a vector u in U with v in V by writing u⊗v, which is called a "pure tensor". This is "bilinear", in that it is additive in each coordinate, and cu⊗v = c(u⊗v) = u⊗cv. One way to construct the tensor product is by taking a bases of U and V, and declaring u_i⊗v_j to be a basis of U⊗V. Alternatively, a "basis-free" construction involves taking the "free vector space" over with basis UxV, and then take the quotient of the space spanned by the "null sums" -- expressions that are forced to be zero by bilinearity. We show why these two constructions are equivalent. Then, we discuss the universal property of the tensor product, which roughly says that U⊗V is the "largest" quotient of a bilinear map from UxV. Then, we see how pure tensors can be represented by rank-1 matrices, and so U⊗V is isomorphic to Hom(U',V), and hence the space of m-by-n matrices. Finally, we see how the tensor product naturally arises behind the scenes anytime we want to extend a real vector space to a complex vector space, which is something that we do without realizing it, whenever we find an eigenvector corresponding to a non-real eigenvalue.

[jongraham]

This is really enlightening... I've always been mystified by the way the tensor product is derived algebraically. The light is starting to come on. Thanks!

[teretx]

Very nice! By far the best video I have seen so far for making the tensor product of vector spaces finally intelligible. Many thanks!

[dacianbonta]

Section 3, Professor, is a beauty! Thank you so much!

[DrMcCrady]

This video helped me a ton! It was nice to see some examples (I sound like one of my students). Now I’m understanding more of Bishop and Goldberg’s book on tensors and manifolds. Thank you!

[Jkfgjfgjfkjg]

@27:00 The composition should be g o q.

[Jkfgjfgjfkjg]

@1:00, you have b1_x instead of b1_y.

[fsaldan]

On 42'49 you cannot multiply a 1xn matrix and an mxn matrix.

[Jkfgjfgjfkjg]

@11:00 How do you prove that x + y cannot be written as u \otimes v for any choice of u in U and v in V?