One cannot fail to notice that in forming linear combinations of linear equations there is no need to continue writing the 'unknowns' x1,...,x, 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 -[九州] 1-0 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 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 &, 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 V be an inner product space, and let W be a subspace of V. The orthogonal complement of W, denoted by W+, is the set of all vectors in V that are orthogonal to every vector in W. Tasks: 1. Prove that Wis a subspace of V. Use the properties of inner products to show that Wis closed under addition and scalar multiplication. 2. Show that if V is finite-dimensional, then V=WW.Provide a detailed proof using the concept of orthogonal projections and the projection theorem, which states that any vector in V can be uniquely decomposed as v=w+w, where we W and w- € W-. 3. Graphically illustrate the concept of orthogonal complement in R² or R³. Use a diagram to represent a plane W and its orthogonal complement, showing how vectors are decomposed into components along W and W- 4. Discuss the significance of orthogonal complements in solving linear systems. Prove that for a matrix A, the null space of A and the row space of A are orthogonal complements in IR", and explain how this property is useful in analyzing solutions to Ax = b.