91Ó°ÊÓ

Proof Let \(A\) be a fixed \(2 \times 3\) matrix. Prove that the set \(W=\left\\{\mathbf{x} \in R^{3}: A \mathbf{x}=\left[\begin{array}{l}1 \\\ 2\end{array}\right]\right\\}\) is not a subspace of \(R^{3}\).

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. \(S=\\{(3,4),(-1,1),(2,0)\\}\)

Is it possible for a transition matrix to equal the identity matrix? Explain.

Proof Let \(A\) be a fixed \(m \times n\) matrix. Prove that the set \(W=\left\\{\mathbf{x} \in R^{n}: A \mathbf{x}=\mathbf{0}\right\\}\) is a subspace of \(R^{n}\).

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. \(S=\\{(2,4),(-1,-2),(0,6)\\}\)

Let \(B\) and \(B^{\prime}\) be two bases for \(R^{n}\). (a) When \(B=I_{n},\) write the transition matrix from \(B\) to \(B^{\prime}\) in terms of \(B^{\prime}\). (b) When \(B^{\prime}=I_{n},\) write the transition matrix from \(B\) to \(B^{\prime}\) in terms of \(B\). (c) When \(B=I_{n},\) write the transition matrix from \(B^{\prime}\) to \(B\) in terms of \(B^{\prime}\). (d) When \(B^{\prime}=I_{m^{\prime}}\) write the transition matrix from \(B^{\prime}\) to \(B\) in terms of \(B\).

Identify and sketch the graph of the conic section. $$ 4 y^{2}-2 x^{2}-4 y-8 x-15=0 $$

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 \(A x=0\) $$ \begin{array}{l} 3 w-2 x+16 y-2 z=-7 \\ -w+5 x-14 y+18 z=29 \\ 3 w-x+14 y+2 z=1 \end{array} $$

The zero vector \(0=(0,0,0)\) can be written as a linear combination of the vectors \(v_{1}, v_{2}\), and \(v_{3}\) because \(0=0 v_{1}+0 v_{2}+0 v_{x}\) This is the trivial solution. Find a nontrivial way of writing 0 as a linear combination of the three vectors, if possible. $$\mathbf{v}_{1}=(1,0,1), \quad \mathbf{v}_{2}=(-1,1,2), \quad \mathbf{v}_{3}=(0,1,4)$$

Identify and sketch the graph of the conic section. $$ x^{2}+4 y^{2}+4 x+32 y+64=0 $$