One cannot fail to notice that in forming linear combinations of linear equations there is no need to continue writing the 'unknowns' x1,...,xn, since one actually computes only with the coefficients A., and the scalars y.. We shall now abbreviate the system (1-1) by where X A AX = Y -[] AL and Y = We call A the matrix of coefficients of the system. Strictly speaking, the rectangular array displayed above is not a matrix, but is a repre- sentation of a matrix. An m X n matrix over the field F is a function A from the set of pairs of integers (i,j), 1≤i≤m, 1≤ j ≤ n, into the field F. The entries of the matrix A are the scalars A(i, j) = Aij, and quite often it is most convenient to describe the matrix by displaying its entries in a rectangular array having m rows and n columns, as above. Thus X (above) is, or defines, an n XI matrix and Y is an m x 1 matrix. For the time being, AX = Y is nothing more than a shorthand notation for our system of linear equations. Later, when we have defined a multi- plication for matrices, it will mean that Y is the product of A and X. We wish now to consider operations on the rows of the matrix A which correspond to forming linear combinations of the equations in the system AX Y. We restrict our attention to three elementary row operations on an m X n matrix A over the field F: 1. multiplication of one row of A by a non-zero scalar e; 2. replacement of the rth row of A by row r plus c times row a, c any scalar and rs; 3. interchange of two rows of A. Do not solve using AI, I want real solutions with graphs and codes, wherever required. Reference is given, if you need further use hoffmann book of LA or maybe Friedberg. Statement: Let A be a matrix of size m x n. The row space and column space of a matrix are subspaces of R" and IR", respectively. Tasks: 1. Prove that the row space of A is the same as the row space of the row echelon form of A. Explain why performing row operations does not change the row space. Use a theoretical argument based on the properties of linear combinations and span. 2. Demonstrate that the dimension of the row space of A is equal to the dimension of the column space of A (which is the rank of A). Provide a general proof using the concepts of linear independence, span, and pivot positions. 3. Explain graphically how the transformation from matrix A to its row echelon form affects the row vectors in terms of their positions in R. Use a diagram to represent the row space before and after the transformation. 4. Discuss why the column space of a matrix A may not be preserved under row operations. Provide a theoretical explanation and support your reasoning with examples.