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 X AX = Y Au AL A Aml -A Y1 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, 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 AXY. We restrict our attention to three elementary row operations on an m Xn 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 8, c any scalar and rs; 3. interchange of two rows of A. Problem 10: Diagonalization and Jordan Canonical Form Statement: A matrix A is said to be diagonalizable if there exists a matrix P such that P-¹AP is a diagonal matrix. The Jordan canonical form generalizes this concept for matrices that are not diagonalizable. Tasks: 1. Prove that a matrix is diagonalizable if and only if the sum of the geometric multiplicities of its eigenvalues equals n, the order of the matrix. Use the concepts of eigenvectors, eigenspaces, and linear independence in your proof. 2. Show that if a matrix is not diagonalizable, it can still be brought into a Jordan canonical form. Explain why this form is the closest one can get to a diagonal matrix and how it represents the structure of the linear transformation. 3. Discuss the relationship between the algebraic multiplicity and geometric multiplicity of eigenvalues in determining whether a matrix is diagonalizable. Provide examples of matrices where the algebraic multiplicity equals the geometric multiplicity and where it does not. 4. Graphically interpret the Jordan canonical form as a combination of scaling and nilpotent transformations. Use diagrams to illustrate how the Jordan blocks correspond to different types of linear transformations, such as scaling along one direction and shifting along another.