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 upquestion:
"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 nontrivial 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 ndimensional vector space V, then there exists a nonzero 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^21, 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...
Homework Statement
The matrix A =(1,2,3;4 5 6) defines a linear transformation T: R^3>R^2 . Find the transformation matrix for T with respect to the basis (1,0,1),(0,2,0),(1,0,1) for R^3 and the basis (0,1),(1,0) for R^2.
Homework Equations

The Attempt at a Solution
I have no...
I'm having some trouble understanding basis and how they relate to transformation matrices.
Homework Statement
Let e_1,e_2,e_3,e_4 be a basis in a four dimensional vector space V. Suppose that the linear transformation F on V has the matrix representation:
[1 0 2 1;1 2 1 3;1 2 5 5;2 2 1...
owlpride: Hmm, regarding the vectors v3 and v4, should they simply be (0, 0,1,0) and (0,0,0,01) ? I think not, but I can't come up with any better criterion... That they're in the plane?
Homework Statement
Find if possible a linear transformation R^4>R^3 so that the nullspace is [(1,2,3,4),(0,1,2,3)] and the range the solutions to x_1+x_2+x_3=0.
Homework Equations

The Attempt at a Solution
So I thought I should start with trying to find what kind of matrix we...
HallsofIvy:
I'll show you where I got the 0*a from a previous example here. I probably misunderstood something, but just to see what I'm thinking.
Let's sa we want to show that f(x,y) = xy is differentiable at (1,1).
f(1+h,1+k)f(1,1) = (1+h)(1+k)1 = h+k+hk =...
Homework Statement
Hello, I'm trying to grasp the definition of a derivative in several variables, that is, to say if it's differentiable at a point.
My book tells me that a function of two variables if differntiable if:
f(a+h,b+k)f(a,b) = A_1h+A_2k+\sqrt{h^2+k^2}\rho(h,k)
And if \rho goes...