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    Tensor product?

    Hurkyl: I'm grateful, but I can't say I understand your point of the message. I feel really stupid for not understanding this.
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    Tensor product of vector space problems

    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...
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    Tensor product?

    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.
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    Tensor product?

    Landau: Thanks, I'll read it through. I'd love to see some "concrete" examples too if possible.
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    Tensor product?

    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...
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    Dual basis problem. (Linear Algebra)

    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.
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    Dual basis problem. (Linear Algebra)

    Then we have to know that it has a non-trivial solution, right? Is there any way to know that besides reasoning that builds on "matrices"?
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    Dual basis problem. (Linear Algebra)

    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?
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    Dual basis problem. (Linear Algebra)

    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.
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    Dual basis problem. (Linear Algebra)

    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...
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    Dual basis problem. (Linear Algebra)

    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...
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    Triple integrals.

    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...
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    Linear transformation given a nullspace and a solution space.

    When I do as we said, I get the matrix: 3 -1 2 -5 2 -3 1 -1 0 1 0 1 But my matrix should be the transpose of that. Anyone willing to help?
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    Another linear algebra problem, basis and linear transformations.

    Ah, it transforms it. I don't see where this is leading me, however :/
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    Another linear algebra problem, basis and linear transformations.

    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...
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    Matrices and bases

    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...
  17. G

    Linear transformation given a nullspace and a solution space.

    Is there any general rule regarding which way to multiplcate composed matrices like this? :) Thanks for all your help!
  18. G

    Linear transformation given a nullspace and a solution space.

    Ah, how come we take it the otherway around though? And why the (1,2,3,4)? :)
  19. G

    Linear transformation given a nullspace and a solution space.

    Call the matrix that we found last for B, the one that we found first for A. Then: inv(B)*A would give it, no?
  20. G

    Linear transformation given a nullspace and a solution space.

    OK, so I have to find the product of the inverse of (1,2,3,4) (0,1,2,3) ,(0,0,1,0),(0,0,0,1) and the matrix we found in the beginning? :)
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    Linear transformation given a nullspace and a solution space.

    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?
  22. G

    Linear transformation given a nullspace and a solution space.

    Isn't it simply the latter matrix that does it?
  23. G

    Linear transformation given a nullspace and a solution space.

    yeah, you're right. I guess: (1,-1,0) would be a vector there, as well would (1,0,-1), right?
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    Linear transformation given a nullspace and a solution space.

    owlpride: I do know how to change bases, but I'm not that good at it. How do you mean with the image?
  25. G

    Linear transformation given a nullspace and a solution space.

    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...
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    Definition of a derivative in several variables

    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 =...
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    Definition of a derivative in several variables

    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...
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