LetAandB bem n matrices. Prove or give a counterexample: IfAx = 0 andBx = 0 have the

By solving a system of equations, find the linear combination of the vectors v1 = 1 0 1 , v2 = 0 1 2 , v3 = 2 1 1 that gives b = 3 0 2 .

For each of the following vectors \(\mathbf{b} \in \mathbb{R}^4\), decide whether b is a linear combination of \(\mathbf{v}_1=\left[\begin{array}{r}1 \\ 0 \\ 1 \\ -2\end{array}\right], \mathbf{v}_2=\left[\begin{array}{r}0 \\ -1 \\ 0 \\ 1\end{array}\right]\), and \(\mathbf{v}_3=\left[\begin{array}{r}1 \\ -2 \\ 1 \\ 0\end{array}\right]\). a. \(\mathbf{b}=\left[\begin{array}{l}1 \\ 1 \\ 1 \\ 1\end{array}\right]\) b. \(\mathbf{b}=\left[\begin{array}{r}1 \\ -1 \\ 1 \\ -1\end{array}\right]\) c. \(\mathbf{b}=\left[\begin{array}{r}1 \\ 1 \\ 0 \\ -2\end{array}\right]\)

Find constraint equations (if any) that b must satisfy in order forAx = b to be consistent. a. A = 3 1 6 2 9 3 b. A = 1 1 1 1 1 2 1 3 4 c. A = 0 1 1 1 2 1 2 1 1 d. A = 1 1 2 2 1 2 1 2 1 e. A = 1 2 1 0 1 1 1 3 4 2 1 1 f. A = 1 1 1 1 1 1 1 1 1 1 2 3

Find constraint equations that b must satisfy in order to be an element of a. V = Span _ (1, 2, 1), (2,4,2) _ b. V = Span _ (1, 0, 1, 1), (0, 1, 1, 2), (1, 1, 1, 0) _ c. V = Span _ (1, 0, 1, 1), (0, 1, 1, 2), (2,1, 1, 0) _ d. V = Span _ (1, 2, 3), (1, 0,2), (1,2, 1) _

By finding appropriate constraint equations, give a Cartesian equation of each of the following planes in \(\mathbb{R}^3\). a. \(\mathbf{x}=s(1,-2,-2)+t(2,0,-1), s, t \in \mathbb{R}\) b. \(\mathbf{x}=(1,2,3)+s(1,-2,-2)+t(2,0,-1), s, t \in \mathbb{R}\) c. \(\mathbf{x}=(4,2,1)+s(1,0,1)+t(1,2,-1), s, t \in \mathbb{R}\)

Suppose A is a 3 4 matrix satisfying the equations A 1 2 1 4 = 1 2 3 and A 0 3 1 2 = 1 1 1 . Find a vector x R4 such thatAx = 0 1 2 . Give your reasoning. (Hint: Look carefully at the vectors on the right-hand side of the equations.)

Find a matrix A with the given property or explain why none can exist. a. One of the rows of A is (1, 0, 1), and for some b R2 both the vectors 1 0 1 and 2 1 1 are solutions of the equation Ax = b. b. The rows of A are linear combinations of (0, 1, 0, 1) and (0, 0, 1, 1), and for some b R2 both the vectors 1 2 1 2 and 4 1 0 3 are solution of the equation Ax = b. c. The rows of A are orthogonal to 1 0 1 0 , and for some nonzero vector b R2 both the vectors 1 0 1 0 and 1 1 1 1 are solutions of the equation Ax = b. d. For some vectors b1, b2 R2 both the vectors 1 0 1 and 2 1 1 are solutions of the equation Ax = b1, and both the vectors 1 0 0 and 1 1 1 are solutions of the equation Ax = b2.

Let \(A=\left[\begin{array}{cc} 1 & \alpha \\ \alpha & 3 \alpha \end{array}\right]\). a. For which numbers \(\alpha\) will A be singular? b. For all numbers \(\alpha\) not on your list in part a, we can solve Ax = b for every vector \(\mathbf{b} \in \mathbb{R}^{2}\). For each of the numbers ? on your list, give the vectors b for which we can solve Ax = b.

Let A = 1 2 1 1 . a. For which numbers will A be singular? b. For all numbers not on your list in part a, we can solve Ax = b for every vector b R3. For each of the numbers on your list, give the vectors b for which we can solve Ax = b. 1

Let A be an \(m \times n\) matrix. Prove or give a counterexample: If Ax = 0 has only the trivial solution x = 0, then Ax = b always has a unique solution.

LetAandB bem n matrices. Prove or give a counterexample: IfAx = 0 andBx = 0 have the same solutions, then the set of vectors b such that Ax = b is consistent is the same as the set of the vectors b such that Bx = b is consistent.

In each case, give positive integers m and n and an example of an m n matrix A with the stated property, or explain why none can exist. a. Ax = b is inconsistent for every b Rm. b. Ax = b has one solution for every b Rm. c. Ax = b has no solutions for some b Rm and one solution for every other b Rm. d. Ax = b has infinitely many solutions for every b Rm. e. Ax = b is inconsistent for some b Rm and has infinitely many solutions whenever it is consistent. f. There are vectors b1, b2, b3 so that Ax = b1 has no solution, Ax = b2 has exactly one solution, and Ax = b3 has infinitely many solutions. _

Suppose A is an \(m \times n\) matrix with rank m and \(\mathbf{v}_1,..., \mathbf{v}_k \in \ \mathbb {R}^n\) are vectors with Span \((\mathbf{v}_1,..., \mathbf{v}_k) =\mathbb {R}^n\). Prove that Span \((A \mathbf {v}_1,...,A \mathbf{v}_k) = \mathbb{R}^m\).

Let A be an \(m \times n\) matrix with row vectors \(mathbf{A}_1, \ldots, \mathbf{A}_m \in \mathbb{R}^n\). *a. Suppose \(\mathbf{A}_1+\cdots+\mathbf{A}_m=\mathbf{0}\). Deduce that \(\operatorname{rank}(A)<m\). (Hint: Why must there be a row of 0's in the echelon form of A?) b. More generally, suppose there is some linear combination \(c_1 \mathbf{A}_1+\cdots+c_m \mathbf{A}_m=\mathbf{0}\), where some \(c_i \neq 0\). Show that \(\operatorname{rank}(A)<m\).

Let A be an \(m \times n\) matrix with column vectors \(\mathbf{a}_{1, \ldots} \mathbf{a}_{n} \in \mathbb{R}^{m}\). a. Suppose \(\mathbf{a}_{1}+\cdots+\mathbf{a}_{n}=0\). Prove that rank(A) < n. (Hint: Consider solutions of Ax = 0.) b. More generally, suppose there is some linear combination \(c_{1} \mathbf{a}_{1}+\cdots+c_{n} \mathbf{a}_{n}=\mathbf{0}\), where some \(c_{i} \neq 0\). Prove that rank(A) < n.