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 ALA A = And A. 21 Y1 X 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≤i≤ 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 × 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 Xn matrix A over the field F: 1. multiplication of one row of A by a non-zero scalar c; 2. replacement of the rth row of A by row r plus c times row 8, c any scalar and r8; 3. interchange of two rows of A. Statement: Let V be a vector space, and let U and W be subspaces of V. The sum of the subspaces U and W, denoted by U+W, is defined as the set of all vectors that can be written as u+w, where u EU and w W. Tasks: 1. Prove that U+W is a subspace of V. Use the definition of vector subspaces to show that U+W is closed under addition and scalar multiplication. 2. Explain the conditions under which the sum U+W is a direct sum, denoted by U + W. Provide a proof that if Unw= {0}, then every element of U+W can be written uniquely as u+w for some u EU and wЄ W. 3. Graphically interpret the concept of a direct sum in 13. Use a diagram to represent two subspaces U and W, such as two planes intersecting only at the origin, and show how vectors in UW can be decomposed uniquely. 4. Prove that if V is a finite-dimensional vector space and V = UW, then dim(V) = dim(U) + dim(W). Explain why this property is significant in the study of linear transformations and vector space decompositions.