91Ó°ÊÓ

Rank, Nullity, Bases, and Linear Independence In Exercises 41 and \(42,\) use the fact that matrices \(A\) and \(B\) are row-equivalent. (a) Find the rank and nullity of \(A\). (b) Find a basis for the nullspace of \(A\). (c) Find a basis for the row space of \(A\). (d) Find a basis for the column space of \(A\). (e) Determine whether the rows of \(A\) are linearly independent. (f) Let the columns of \(A\) be denoted by \(a_{1}, a_{2}, a_{y}, a_{4},\) and \(a_{5}\) Determine whether each set is linearly independent. (i) \(\left\\{a_{1}, a_{2}, a_{4}\right\\}\) (ii) \(\left\\{a_{1}, a_{2}, a_{3}\right\\}\) (iii) \(\left\\{\mathbf{a}_{1}, \mathbf{a}_{3}, \mathbf{a}_{\mathrm{s}}\right\\}\) $$ A=\left[\begin{array}{rrrrr} 1 & 2 & 1 & 0 & 0 \\ 2 & 5 & 1 & 1 & 0 \\ 3 & 7 & 2 & 2 & -2 \\ 4 & 9 & 3 & -1 & 4 \end{array}\right] $$

In Exercises 41 and \(42,\) use the fact that matrices \(A\) and \(B\) are row- equivalent. (a) Find the rank and nullity of \(A\). (b) Find a basis for the nullspace of \(A\). (c) Find a basis for the row space of \(A\). (d) Find a basis for the column space of \(A\). (e) Determine whether the rows of \(A\) are linearly independent. (f) Let the columns of \(A\) be denoted by \(a_{1}, a_{2}, a_{3}, a_{4},\) and \(a_{5}\) Determine whether each set is linearly independent. (i) \(\left\\{a_{1}, a_{2}, a_{4}\right\\}\) (ii) \(\left\\{a_{1}, a_{2}, a_{3}\right\\}\) (iii) \(\left\\{\mathbf{a}_{1}, \mathbf{a}_{3}, \mathbf{a}_{5}\right\\}\) $$\begin{aligned}&A=\left[\begin{array}{rrrrr}-2 & -5 & 8 & 0 & -17 \\\1 & 3 & -5 & 1 & 5 \\\3 & 11 &-19 & 7 & 1 \\\1 & 7 & -13 & 5 & -3\end{array}\right]\\\&B=\left[\begin{array}{lllll}1 & 0 & 1 & 0 & 1 \\\0 & 1 & -2 & 0 & 3 \\\0 & 0 & 0 & 1 & -5 \\\0 & 0 & 0 & 0 & 0\end{array}\right]\end{aligned}$$

Finding a Basis and Dimension In Exercises \(43-48\), find (a) a basis for and (b) the dimension of the solution space of the homogeneous system of linear equations. $$ \begin{array}{r} x-2 y+3 z=0 \\ -3 x+6 y-9 z=0 \end{array} $$

determine whether the set, together with the standard operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. CAPSTONE A. Describe the conditions under which a set may be classified as a vector space. B. Give an example of a set that is a vector space and an example of a set that is not a vector space.

Determine whether the nonhomogeneous system \(A x=b\) is consistent. If it is, write the solution in the form \(\mathbf{x}=\mathbf{x}_{p}+\mathbf{x}_{\mathbf{k}},\) where \(\mathbf{x}_{\mathbf{p}}\) is a particular solution of \(\mathbf{A} \mathbf{x}=\mathbf{b}\) and \(x_{k}\) is a solution of \(A x=0\) $$ \begin{array}{c} x+2 y-4 z=-1 \\ -3 x-6 y+12 z=3 \end{array} $$

Determine whether the set of vectors in \(M_{2,2}\) is linearly independent or linearly dependent. \(A=\left[\begin{array}{rr}2 & 0 \\ -3 & 1\end{array}\right], B=\left[\begin{array}{rr}-4 & -1 \\ 0 & 5\end{array}\right], C=\left[\begin{array}{rr}-8 & -3 \\ -6 & 17\end{array}\right]\)

Show that the set is linearly dependent by finding a nontrivial linear combination of vectors in the set whose sum is the zero vector. Then express one of the vectors in the set as a linear combination of the other vectors in the set. \(S=\\{(3,4),(-1,1),(2,0)\\}\)

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) If \(P\) is the transition matrix from a basis \(B\) to \(B^{\prime}\), then the equation \(P[\mathbf{x}]_{n^{\prime}}=[\mathbf{x}]_{B}\) represents the change of basis from \(B\) to \(B^{\prime}\). (b) If \(B\) is the standard basis in \(R^{n}\), then the transition matrix from \(B\) to \(B^{\prime}\) is \(P^{-1}=\left(B^{\prime}\right)^{-1}\). (c) For any \(4 \times 1\) matrix \(X\), the coordinate matrix \([X]_{s}\) relative to the standard basis for \(M_{4,1}\) is equal to \(X\) itself.

Determine whether each statement is true or false. If a statement is true, give a reason or cite an appropriate statement from the text. If a statement is false, provide an example that shows the statement is not true in all cases or cite an appropriate statement from the text. (a) If \(P\) is the transition matrix from a basis \(B^{\prime}\) to \(B\) then \(P^{-1}\) is the transition matrix from \(B\) to \(B^{\prime}\). (b) To perform the change of basis from a nonstandard basis \(B^{\prime}\) to the standard basis \(B\), the transition matrix \(P^{-1}\) is simply \(B^{\prime}\). (c) The coordinate matrix of \(p=-3+x+5 x^{2}\) relative to the standard basis for \(P_{2}\) is \([p]_{s}=\left[\begin{array}{lll}5 & 1 & -3\end{array}\right]^{T}\).

In Exercises \(57-62,\) determine whether \(\mathbf{b}\) is in the column space of \(A\). If it is, write \(b\) as a linear combination of the column vectors of \(A\). $$A=\left[\begin{array}{rrr}1 & 3 & 2 \\\\-1 & 1 & 2 \\\0 & 1 & 1\end{array}\right], \quad\mathbf{b}=\left[\begin{array}{l}1 \\\1 \\\0\end{array}\right]$$