Prove that the reduced echelon form of a matrix is unique, as follows. Suppose B and C

Use elementary operations to find the general solution of each of the following systems of equations. Use the method of Example 1 as a prototype. a. x1 + x2 = 1 x1 + 2x2 + x3 = 1 x2 + 2x3 = 1 b. x1 + 2x2 + 3x3 = 1 2x1 + 4x2 + 5x3 = 1 x1 + 2x2 + 2x3 = 0 c. 3x1 6x2 x3 + x4 = 6 x1 + 2x2 + 2x3 + 3x4 = 3 4x1 8x2 3x3 2x4 = 3

Decide which of the following matrices are in echelon form, which are in reduced echelon form, and which are neither. Justify your answers.

For each of the following matrices A, determine its reduced echelon form and give the general solution of Ax = 0 in standard form. a. A = 1 0 1 2 3 1 3 3 0 b. A = 2 2 4 1 1 2 3 3 6 c. A = 1 2 1 1 3 1 2 4 3 1 1 6 d. A = 1 1 1 1 1 2 1 2 1 3 2 4 1 2 2 3 e. A = _ 1 2 1 0 2 4 3 1 _ f. A = 1 2 0 1 1 1 3 1 2 3 1 1 3 1 1 2 3 7 3 4 g. A = 1 1 1 1 0 1 0 2 1 1 0 2 2 2 0 1 1 1 0 1 h. A = 1 1 0 5 0 1 0 1 1 3 2 0 1 2 3 4 1 6 0 4 4 12 1 7

Give the general solution of the equation Ax = b in standard form.

For the following matrices A, give the general solution of the equation \(A \mathbf{x}=\mathbf{x}\) in standard form. (Hint: Rewrite this as \(B \mathbf{x}=\mathbf{0}\) for an appropriate matrix B.) a. \(A=\left[\begin{array}{cc}10 & -6 \\ 18 & -11\end{array}\right]\) b. \(A=\left[\begin{array}{rrr}1 & 0 & 0 \\ -2 & 1 & 2 \\ -2 & 0 & 3\end{array}\right]\) c. \(A=\left[\begin{array}{rrr}0 & -1 & 0 \\ 0 & 0 & -1 \\ 1 & 0 & 0\end{array}\right]\)

For the following matrices A, give the general solution of the equation Ax = 2x in standard form. a. A = _ 0 1 2 1 _ b. A = 3 16 15 1 12 9 1 16 13

One might need to find solutions of Ax = b for several different bs, say b1, . . . , b k . In this event, one can augment the matrix A with all the bs simultaneously, forming the multi-augmented matrix [A | b1 b2 bk ]. One can then read off the various solutions from the reduced echelon form of the multi-augmented matrix. Use this method to solve Ax = bj for the given matrices A and vectors bj . a. A = 1 0 1 2 1 1 1 2 2 , b1 = 1 1 5 , b2 = 1 3 2 b. A = _ 1 2 1 0 2 3 2 1 _ , b1 = _ 1 0 _ , b2 = _ 0 1 _ c. A = 1 1 1 0 1 1 1 2 1 , b1 = 1 0 0 , b2 = 0 1 0 , b3 = 0 0 1

Find all the unit vectors x R3 that make an angle of /3 with each of the vectors (1, 0,1) and (0, 1, 1).

Find all the unit vectors x R3 that make an angle of /4 with (1, 0, 1) and an angle of /3 with (0, 1, 0).

Find a normal vector to the hyperplane in R4 spanned by a. (1, 1, 1, 1), (1, 2, 1, 2), and (1, 3, 2, 4); b. (1, 1, 1, 1), (2, 2, 1, 2), and (1, 3, 2, 3).

Find all vectors x \(\in \mathbb{R}^{2}\) that are orthogonal to both a. (1, 0, 1, 1) and (0, 1, -1, 2); b. (1, 1, 1, -1) and (1, 2, -1, 1).

Find all the unit vectors in R4 that make an angle of /3 with (1, 1, 1, 1) and an angle of /4 with both (1, 1, 0, 0) and (1, 0, 0, 1). _

Let A be an m n matrix, let x, y Rn, and let c be a scalar. Show that a. A(cx) = c(Ax) b. A(x + y) = Ax + Ay

Let A be an m n matrix, and let b Rm. a. Show that if u and v Rn are both solutions of Ax = b, then u v is a solution of Ax = 0. b. Suppose u is a solution of Ax = 0 and p is a solution of Ax = b. Show that u + p is a solution of Ax = b. Hint: Use Exercise 13.

a. Prove or give a counterexample: If A is an \(m \times n\) matrix and \(\mathbf{x} \in \mathbb{R}^n\) satisfies \(A \mathbf{x}=\mathbf{0}\), then either every entry of A is 0 or \(\mathbf{x}=\mathbf{0}\). b. Prove or give a counterexample: If A is an \(m \times n\) matrix and \(A \mathbf{x}=\mathbf{0}\) for every vector \(\mathbf{x} \in \mathbb{R}^n\), then every entry of A is 0 .

Prove that the reduced echelon form of a matrix is unique, as follows. Suppose B and C are reduced echelon forms of a given nonzero \(m \times n\) matrix A. a. Deduce from the proof of Proposition 4.2 that B andC have the same pivot variables. b. Explain why the pivots of B and C are in the identical positions. (This is true even without the assumption that the matrices are in reduced echelon form.) c. By considering the solutions in standard form of Bx = 0 and Cx = 0, deduce that B = C.

In rating the efficiency of different computer algorithms for solving a system of equations, it is usually considered sufficient to compare the number of multiplications required to carry out the algorithm. a. Show that n(n 1) + (n 1)(n 2)+ +(2)(1) = _n k=1 (k2 k) multiplications are required to bring a general n n matrix to echelon form by (forward) Gaussian elimination. b. Show that _n k=1 (k2 k) = 13 (n3 n). (Hint: For some appropriate formulas, see Exercise 1.2.10.) c. Now show that it takes n + (n 1) + (n 2)+ +1 = n(n + 1)/2 multiplications to bring the matrix to reduced echelon form by clearing out the columns above the pivots, working right to left. Show that it therefore takes a total of 13 n3 + 12 n2 + 16 n multiplications to put A in reduced echelon form. d. Gauss-Jordan elimination is a slightly different algorithm used to bring a matrix to reduced echelon form: Here each column is cleared out, both below and above the pivot, before moving on to the next column. Show that in general this procedure requires n2(n 1)/2 multiplications. For large n, which method is preferred?