One cannot fail to notice that in forming linear combinations of linear equations there is no need to continue writing the 'unknowns' x1,..., n, 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 AL A = - Am 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≤ 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 × 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. Problem 4: Orthogonal Projections and Least Squares Solutions Statement: Let V be a vector space and let W be a subspace of V. The orthogonal projection of a vector v onto the subspace W, denoted by projw (v), is defined as the vector in W that is closest to v. Tasks: 1. Prove that the orthogonal projection of a vector v onto a subspace W is unique. Provide a detailed theoretical argument using properties of inner products and the definition of orthogonality. 2. Show that if W is spanned by an orthonormal set of vectors {u, u2,...,us), then the orthogonal projection of v onto W can be written as: proju (v) (vu)u, Explain why this formula holds and use it to derive the least squares solution to the system of equations Ax = b, where A is a matrix with linearly independent columns. 3. Graphically interpret the orthogonal projection in the context of R3, where W is a plane through the origin. Show how a vector is projected onto the plane and explain the role of the orthogonal complement in this process. Use a diagram to illustrate the components of the vector along the plane and perpendicular to it. 4. Discuss the significance of orthogonal projections in solving overdetermined linear systems. Provide a theoretical argument for why the least squares solution minimizes the error vector ||Ax - b.