Tensor product of vector space problms
Homework Statement
I'm currently reading Halmos's book "Finite dimensional vector spaces" and I find it excellent. However, I'm having some problems with his definition of the tensor product of two vector spaces, and I hope you could help me clear it...
Some notation in there that I'm quite not used to. What's really confusing me are some concrete examples, say, how to use (any given) definitions to form the tensor product of say, C^2 and C^3.
Hello,
I'm currently reading Halmos's book "Finite dimensional vector spaces" and I find it excellent. However, I'm having some problems with his definition of the tensor product of two vector spaces, and I hope you could help me clear it out. Here's what he writes:
"Definition: The tensor...
And oh, there's a small follow up-question:
"What does this result say in general about the solutions of linear equations?"
I'd say that if we have n unknowns and m equations, there's a non-trivial solution, but that's just me.
Tedjn:
Linear systems! Then if each y_j(e_i) produces a constant real number, we have a system of linear equations in n unknowns with m equations, right?
Tedjn: I've used the properties to rewrite them into the form:
a_1y_1(e_1)+...+a_ny_1(e_n) = 0
etc. for all functionals. But still, nothing.
Office_Shredder:
I don't really understand what you're hinting at, sorry.
I've reached that point before, but from there I'm kinda stuck. My first thought was letting the first m entries of the vector be 0, but that wouldn't do anything. I thought something about representing each linear functional as a linear combination of the dual base vectors, but well, not...
Homework Statement
Prove that if m < n and if y_1,...,y_m are linear functionals on an n-dimensional vector space V, then there exists a non-zero vector x in V such that [x,y_j] = 0 for j = 1,..., m
Homework Equations
The Attempt at a Solution
My thinking is somehow that we...
Homework Statement
Integrate
\int_D z dxdydz where D is z\geq 0, z^2*\geq 2x^2+3y^2-1, x^2+y^2+z^2 \leq 3
Homework Equations
Spherical coordinates? I'm stuck. I have problems finding the boundaries of integration.
The Attempt at a Solution
None. I'd be most grateful for help...