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Image of a Matrix and symmetric matrix

  1. Feb 4, 2015 #1
    Hi,
    Well I hope it's not a thread that already is in the storage here. I want to understand the image of a matrix. not only calculating it but also why I'm doing that. Here are my questions:

    1) They say d = Lx has a solution if d ∈ ImL. I know that the image of a matrix is calculated by Lx = x. but here we have d = Lx.

    2) Can anybody show me what image of a matrix and what solution the question above has geometrically? I mean by firgures?

    3) If A is a nxn matrix and B is a nx1 matrix and you only have the image of A , what does it say above the image of Γ = [B, AB]?

    4) If We know the image of A and B where both A and B are nxn. what do you know about the image of C=AB , C = A+B?

    and last question which doesnt have so much to do with image:
    5) When is the product of two matrices symmetric? by only looking at them?

    I appritiate all kind of help. thanks in advance
     
    Last edited: Feb 4, 2015
  2. jcsd
  3. Feb 4, 2015 #2
    The image (range, column space) of a matrix is not calculated by ##L\mathbf x= \mathbf x##, check your source.
     
  4. Feb 4, 2015 #3

    Svein

    User Avatar
    Science Advisor

    I am not familiar with your terminology, but I will make some statements that I hope are relevant.
    1. A matrix is a representation of a linear operator in ℝn
    2. Therefore matrices obeys the rules for linear operators - if A is a matrix, X and Y are points in ℝn and α, β are real numbers, then A(αX+βY) = αAX + βAY.
    3. Matrix multiplication is defined for certain combinations of matrices - if we have Am,p and Bp,n, then Cm,n = A⋅B exists.
    4. Matrix multiplication is not commutative - in general A⋅B ≠ B⋅A
    A linear operator is symmetric with respect to an inner product <,> if <AX,y> = <x,Ay>
     
  5. Feb 4, 2015 #4

    Mark44

    Staff: Mentor

    A matrix is a representation of a linear transformation, which you can think of as a kind of function. The image of a linear transformation is the set of possible outputs from that transformation.

    In another reply, Svein called this a linear operator, but as I recall things, a linear operator is a transformation (or mapping) from a given vector space to itself. Hence any matrix for a linear operator has to be square. For a linear transformation, the dimensions of the domain and range don't have to be the same. Hence, a matrix for such a linear transformation can by m x n, where m and n aren't equal. In this case, the linear transformation maps vectors in Rn (the domain) to vectors in Rm (the range).
    The image is not calculated by Lx = x. A vector y is in the image of L -- Im(L) -- if and only if there exists a vector x in the domain of L such that Lx = y.
    Here's a simple example where L is a transformation of R2 to R2, with this matrix:
    $$A = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix}$$
    A maps a vector <x, y> to <y, 0>. (Check this for yourself.) Geometrically, A reflects an input vector across the line y = x, and then projects that vector onto the x-axis. These two steps transform <x, y> to <y, x> and then to <y, 0>. So for example,
    $$A \begin{bmatrix}2 \\ 3 \end{bmatrix} = \begin{bmatrix} 0 & 1 \\ 0 & 0 \end{bmatrix} \begin{bmatrix}2 \\ 3 \end{bmatrix} = \begin{bmatrix}3 \\ 0 \end{bmatrix}$$

    Another piece of information that is related to the image of a transformation is its nullspace, the subspace of the domain that is mapped to the zero vector in the range. A vector x is in the nullspace of L iff Lx = 0. As it turns out, the nullspace of the matrix of my example is the set of vectors in R2 whose y-component is 0; in other words, any vector that lies along the x-axis.
    I don't know what this notation means.
     
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