Can you use elementary column operations (same as the elementary row operations,

Question

Can you use elementary column operations (same as the elementary row operations, but on the
columns instead) to solve a system of equations? Explain. An example might be helpful here...
(3) As shown in class, matrix multiplication is not commutative. However, there are still some n x n matrices, B, that commute with all other n x n matrices (that is AB = BA for any n x n matrix A). Find all such B for 2 x 2 matrices.
(4) Determine if the following functions are linear. If so, find the matrix representing the transformation.
(i) T(a, b) = (a + b; b - 2a)
(ii)T(a, b) = (a, b, ab)
(iii)T(a, b, c) = (c, b, a)
(iv) Fix a vector v in R3. T = R3 --> R defined by T( w) = v * w
(v) Fix a vector v in R3. T : R3 --> R3 defined by T( w) = v * w
(5) Consider a square with corners (1, 1), (-1, 1), (-1,-1), and (1,-1) in R2. Find all linear transformations of the plane (in matrix form) that transform the square to itself.

Explanation / Answer

You may not ask more than one question when you post asking for help. Therefore, I'll only answer your first question. Yes, you can use elementary column operations to solve a system of equations. Take the transpose of a matrix, then use elementary row operations, then take the transpose again. You will see that this is equivalent to using elementary column operations.

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