Question

Lecture 11
Confused,
Strang says dim 3 by 3 symmetric subspace is 6. I get only 3. Assuming he means A transpose=A. Is he using this definition?

Answer 1

Yes, I think he's using that definition. He's talking about the dimension of the space of symmetric 3x3 matrices. I think 6 is right. Let's say you're constructing a symmetric 3x3 matrix. You will need to write down 9 numbers. How many of the 9 numbers can you choose freely? You can't choose all 9 freely because of the requirement that the matrix be symmetic. The number of numbers that you can choose freely is the dimension of the space.

Answer 2

That makes sense.
He is then referring to the set of all symmetric 3x3's; not the set of all rank one matrices that are symmetric. Lifting that assumption means the basis set of all symmetric matrices may be of rank one or two. A 3x3 matrix of rank 3 would be a linear combination of two of the basis matrices. Does that sound right?

Answer 3

yes, i think so

Answer 4

basises are composed of vectors, not of sets. the dimension of a subspace is the number of basis vectors needed to span it. you can't say "a set of rank two matrices" as a basis vector -- you would need to specify a particular matrix. You need 6 basis matrices to span the subspace of symmetric 3x3 matrices. These basis matrices are the 3 with a 1 on the diagonal and the 3 symmetric matrices with a 1 in each mirrored spot above and below the diagonal. A 3x3 matrix can be described with an infinite variety of linear combinations, but the space of all 3x3 symmetric matrices can be spanned entirely by a finite number of independent basis matrices which can be combined to create every possible symmetric matrix.