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 ALM A = Amm NJ- 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 X I 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 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 8, c any scalar and rs; 3. interchange of two rows of A. Statement: Let T: V→ V be a linear transformation on a vector space V. A subspace U CV is called invariant under T if T(u) = U for all u EU. Tasks: 1. Prove that the set of all invariant subspaces under a linear transformation T forms a lattice. Explain the properties of this lattice and how the operations of intersection and sum of invariant subspaces relate to it. 2. Discuss the conditions under which an invariant subspace can be decomposed into a direct sum of smaller invariant subspaces. Provide a theoretical argument involving eigenvalues, eigenvectors, and generalized eigenvectors of T. 3. Graphically interpret the concept of invariant subspaces in R3. For example, if I represents a rotation about an axis, show how the subspace spanned by the rotation axis is invariant. Use diagrams to illustrate invariant lines or planes. 4. Explain why the existence of non-trivial invariant subspaces is important for diagonalizing a matrix or finding its Jordan form. Provide a proof that if all eigenvalues of a matrix are distinct, then the matrix has a complete set of linearly independent eigenvectors, leading to diagonalizability.

Intermediate Algebra

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