Image of a Matrix and symmetric matrix

[Payam30]

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 doesn't 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

 

[Hawkeye18]

The image (range, column space) of a matrix is not calculated by $L\mathbf x= \mathbf x$, check your source.

 

[Svein]

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 A_m,p and B_p,n, then C_m,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>

 

[Mark44]

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.

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 R^m (the range).

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.

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.

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

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.

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]?

I don't know what this notation means.

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 doesn't 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