One cannot fail to notice that in forming linear combinations of linear equations there is no need to continue writing the 'unknowns' x1,...,, since one actually computes only with the coefficients A., and the scalars y. We shall now abbreviate the system (1-1) by 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. where AX = Y An ALN A Aml Am N- N-- 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) = A., 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 I matrix. For the time being, AXY 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 rows, c any scalar and rs; 3. interchange of two rows of A. Problem 5: Eigenvalues, Eigenvectors, and Diagonalization Statement: A matrix A is said to be diagonalizable if there exists a matrix P such that P-¹AP = D. where D is a diagonal matrix consisting of the eigenvalues of A. Tasks: 1. Prove that if a matrix A has n linearly independent eigenvectors, then it is diagonalizable. Provide a step-by-step proof that shows how the eigenvectors form a basis for R.", and explain why this leads to diagonalization. 2. Discuss the conditions under which a matrix is not diagonalizable. Provide a theoretical explanation of why having fewer than na linearly independent eigenvectors prevents diagonalization. Use an example of a defective matrix to illustrate this concept. 3. Graphically interpret the diagonalization process as a change of basis in 12". Explain how the original matrix transformation becomes a scaling transformation in the new basis formed by the eigenvectors. Use diagrams to show how vectors are transformed under both the original and the diagonalized representations. 4. Explain the significance of diagonalization in simplifying the computation of powers of a matrix. A*, where k is a positive integer. Prove that if A is diagonalizable, then A = PD*P-1, and discuss the computational advantages of this approach.