In Exercises 17 and 18, mark each statement True or False.

Problem 30E [M] Let H = Span {v1, v2, v3} and B = {v1, v2, v3}. Showthat is a basis for H and x is in H, and find the -coordinate vector of x, when

Problem 1E In Exercises 1 and 2, find the vector x determined by the given coordinate vector and the given basis . Illustrate your answer with a figure, as in the solution of Practice Problem 2.

Problem 2E In Exercises, find the vector x determined by the given coordinate vector [x] and the given basis . Illustrate your answer with a figure, as in the solution of Practice Problem. Practice Problem Consider the basis

Problem 3E In Exercises, the vector x is in a subspace H with a basis = {b1, b2}. Find the -coordinate vector of x.

Problem 4E In Exercises, the vector x is in a subspace H with a basis = {b1, b2}. Find the -coordinate vector of x.

Problem 6E In Exercises, the vector x is in a subspace H with a basis = {b1, b2}. Find the -coordinate vector of x.

Problem 5E In Exercises, the vector x is in a subspace H with a basis = {b1, b2}. Find the -coordinate vector of x.

Exercises 9–12 display a matrix A and an echelon form of A. Find bases for Col A and Nul A, and then state the dimensions of these subspaces. \(\begin{aligned}A &=\left[\begin{array}{rrrrr}1 & -2 & 9 & 5 & 4 \\ 1 & -1 & 6 & 5 & -3 \\ -2 & 0 & -6 & 1 & -2 \\ 4 & 1 & 9 & 1 & -9\end{array}\right] \\ & \sim\left[\begin{array}{rrrrr}1 & -2 & 9 & 5 & 4 \\ 0 & 1 & -3 & 0 & -7 \\ 0 & 0 & 0 & 1 & -2 \\ 0 & 0 & 0 & 0 & 0\end{array}\right]\end{aligned}\)

Problem 9E Exercises display a matrix A and an echelon form of A. Find bases for Col A and Nul A, and then state the dimensions of these subspaces.

Let \(\mathbf{b}_1=\left[\begin{array}{l}0 \\ 2\end{array}\right], \mathbf{b}_2=\left[\begin{array}{l}2 \\ 1\end{array}\right], \mathbf{x}=\left[\begin{array}{r}-2 \\ 3\end{array}\right], \mathbf{y}=\left[\begin{array}{l}2 \\ 4\end{array}\right]\), \(\mathbf{z}=\left[\begin{array}{r}-1 \\ -2.5\end{array}\right]\), and \(\mathcal{B}=\left\{\mathbf{b}_1, \mathbf{b}_2\right\}\). Use the figure to estimate \([\mathbf{x}]_{\mathcal{B}},[\mathbf{y}]_{\mathcal{B}}\), and \([\mathbf{z}]_{\mathcal{B}}\). Confirm your estimates of \([\mathbf{y}]_{\mathcal{B}}\) and \([\mathbf{z}]_{\mathcal{B}}\) by using them and \(\left\{\mathbf{b}_1, \mathbf{b}_2\right\}\) to compute \(\mathbf{y}\) and \(\mathbf{z}\).

Problem 7E Use the figure to estimate Confirm your estimate of by using it and to compute x.

Problem 11E Exercises display a matrix A and an echelon form of A. Find bases for Col A and Nul A, and then state the dimensions of these subspaces.

Problem 13E In Exercises 13 and 14, find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace?

In Exercises 13 and 14, find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace? \(\left[\begin{array}{r}1 \\ -1 \\ -2 \\ 5\end{array}\right],\left[\begin{array}{r}2 \\ -3 \\ -1 \\ 6\end{array}\right],\left[\begin{array}{r}0 \\ 2 \\ -6 \\ 8\end{array}\right],\left[\begin{array}{r}-1 \\ 4 \\ -7 \\ 7\end{array}\right],\left[\begin{array}{r}3 \\ -8 \\ 9 \\ -5\end{array}\right]\)

Exercises 9–12 display a matrix A and an echelon form of A. Find bases for Col A and Nul A, and then state the dimensions of these subspaces. \(A=\left[\begin{array}{rrrrr} 1 & 2 & -4 & 3 & 3 \\ 5 & 10 & -9 & -7 & 8 \\ 4 & 8 & -9 & -2 & 7 \\ -2 & -4 & 5 & 0 & -6 \end{array}\right]\) \(\sim\left[\begin{array}{rrrrr} 1 & 2 & -4 & 3 & 3 \\ 0 & 0 & 1 & -2 & 0 \\ 0 & 0 & 0 & 0 & -5 \\ 0 & 0 & 0 & 0 & 0 \end{array}\right]\)

Suppose a \(4 \times 7\) matrix A has three pivot columns. Is Col \(A=\mathbb{R}^{3}\)? What is the dimension of Nul A? Explain your answers.

Problem 15E Suppose a 3 × 5 matrix A has three pivot columns. Is Col A = ?3? Is Nul A = ?2? Explain your answers.

Problem 19E In Exercises 19–24, justify each answer or construction. If the subspace of all solutions of Ax = 0 has a basis consisting of three vectors and if A is a 5 × 7 matrix, what is the rank of A?

In Exercises 17 and 18, mark each statement True or False. Justify each answer. Here A is an \(m \times n\) matrix. a. If \(\mathcal{B}\) is a basis for a subspace H, then each vector in H can be written in only one way as a linear combination of the vectors in \(\mathcal{B}\). b. If \(\mathcal{B}=\left\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{p}\right\}\) is a basis for a subspace H of \(\mathbb{R}^{n}\), then the correspondence \(\mathbf{x} \mapsto[\mathbf{x}]_{\mathcal{B}}\) makes H look and act the same as \(\mathbb{R}^{p}\). c. The dimension of Nul A is the number of variables in the equation Ax = 0. d. The dimension of the column space of A is rank A. e. If H is a p-dimensional subspace of \(\mathbb{R}^{n}\), then a linearly independent set of p vectors in H is a basis for H.

Problem 20E In Exercises, justify each answer or construction. What is the rank of a 4 × 5 matrix whose null space is three-dimensional?

Problem 22E In Exercises 19–24, justify each answer or construction. Show that a set in is linearly dependent if dim Span .

Problem 21E In Exercises, justify each answer or construction. If the rank of a 7 × 6 matrix A is 4, what is the dimension of the solution space of Ax = 0?

In Exercises 19–24, justify each answer or construction. Construct a \(4 \times 3\) matrix with rank l.

Problem 23E In Exercises, justify each answer or construction. If possible, construct a 3 × 4 matrix A such that dim Nul A = 2 and dim Col A = 2.

Problem 26E Suppose columns 1, 3, 5, and 6 of a matrix A are linearly independent (but are not necessarily pivot columns) and the rank of A is 4. Explain why the four columns mentioned must be a basis for the column space of A.

Problem 25E Let A be an n × p matrix whose column space is p dimensional. Explain why the columns of A must be linearly independent.

Problem 27E Suppose vectors span a subspace W, and let be any set in W containing more than p vectors. Fill in the details of the following argument to show that a. Explain why for each vector aj , there exists a vector cj in such that . b. Let Explain why there is a nonzero vector u such that c. Use B and C to show that Au = 0: This shows that the columns of A are linearly dependent.

Use Exercise 27 to show that if \(\mathcal{A} \text { and } \mathcal{B}\) are bases for a subspace W of \(\mathbb{R}^{n}\), then A cannot contain more vectors than \(\mathcal{B}\), and, conversely, \(\mathcal{B}\) cannot contain more vectors than \(\mathcal{A}\).

Problem 29E [M] Let H = Span {v1, v2} and = {v1, v2}. Show that x is in H, and find the -coordinate vector of x, when

Problem 17E Suppose a 4 × 7 matrix A has three pivot columns. Is ? What is the dimension of Nul A? Explain your answers.