91Ó°ÊÓ

Let \(V\) be the first quadrant in the \(x y\) -plane; that is, let \(V=\left\\{\left[\begin{array}{l}{x} \\ {y}\end{array}\right] : x \geq 0, y \geq 0\right\\}\) a. If \(\mathbf{u}\) and \(\mathbf{v}\) are in \(V,\) is \(\mathbf{u}+\mathbf{v}\) in \(V ?\) Why? b. Find a specific vector \(\mathbf{u}\) in \(V\) and a specific scalar \(c\) such that \(c \mathbf{u}\) is \(n o t\) in \(V .\) (This is enough to show that \(V\) is not a vector space.)

For each subspace in Exercises 1–8, (a) find a basis, and (b) state the dimension. $$ \left\\{\left[\begin{array}{c}{s-2 t} \\ {s+t} \\ {3 t}\end{array}\right] : s, t \text { in } \mathbb{R}\right\\} $$

Let \(H\) be the set of points inside and on the unit circle in the \(x y\) -plane. That is, let \(H=\left\\{\left[\begin{array}{c}{x} \\\ {y}\end{array}\right] : x^{2}+y^{2} \leq 1\right\\} .\) Find a specific example - two vectors or a vector and a scalar - to show that \(H\) is not a subspace of \(\mathbb{R}^{2}\) .

In Exercises \(5-8,\) find the coordinate vector \([\mathbf{x}]_{\mathcal{B}}\) of \(\mathbf{x}\) relative to the given basis \(\mathcal{B}=\left\\{\mathbf{b}_{1}, \ldots, \mathbf{b}_{n}\right\\}\) $$ \mathbf{b}_{1}=\left[\begin{array}{r}{1} \\ {-2}\end{array}\right], \mathbf{b}_{2}=\left[\begin{array}{r}{5} \\ {-6}\end{array}\right], \mathbf{x}=\left[\begin{array}{l}{4} \\ {0}\end{array}\right] $$

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

In Exercises \(7-14\) , cither use an appropriate theorem to show that the given set, \(W,\) is a vector space, or find a specific example to the contrary. $$ \left\\{\left[\begin{array}{c}{c-6 d} \\ {d} \\ {c}\end{array}\right] : c, d \text { real }\right\\} $$

Determine the dimensions of Nul \(A\) and \(\mathrm{Col} A\) for the matrices shown in Exercises \(13-18\) . $$ A=\left[\begin{array}{rrrrrr}{1} & {3} & {-4} & {2} & {-1} & {6} \\ {0} & {0} & {1} & {-3} & {7} & {0} \\ {0} & {0} & {0} & {1} & {4} & {-3} \\ {0} & {0} & {0} & {0} & {0} & {0}\end{array}\right] $$

In Exercises \(15-18,\) find a basis for the space spanned by the given vectors, \(\mathbf{v}_{1}, \ldots, \mathbf{v}_{5}\) $$ [\mathbf{M}]\left[\begin{array}{r}{-8} \\ {7} \\ {6} \\ {5} \\\ {-7}\end{array}\right],\left[\begin{array}{r}{8} \\ {-7} \\ {-9} \\ {-5} \\\ {-5} \\ {7}\end{array}\right],\left[\begin{array}{r}{-8} \\ {7} \\ {4} \\\ {5} \\ {-7}\end{array}\right],\left[\begin{array}{r}{1} \\ {4} \\ {9} \\\ {6} \\ {-7}\end{array}\right],\left[\begin{array}{r}{-9} \\ {4} \\ {9} \\\ {-1} \\ {0}\end{array}\right] $$

For the matrices in Exercises \(17-20,(a)\) find \(k\) such that Nul \(A\) is a subspace of \(\mathbb{R}^{k},\) and \((b)\) find \(k\) such that \(\operatorname{Col} A\) is a subspace of \(\mathbb{R}^{k} .\) $$ A=\left[\begin{array}{rrrrr}{4} & {5} & {-2} & {6} & {0} \\ {1} & {1} & {0} & {1} & {0}\end{array}\right] $$

In Exercises 25 and \(26, A\) denotes an \(m \times n\) matrix. Mark each statement True or False. Justify each answer. a. The null space of \(A\) is the solution set of the equation \(A \mathbf{x}=\mathbf{0} .\) b. The null space of an \(m \times n\) matrix is in \(\mathbb{R}^{m}\) . c. The column space of \(A\) is the range of the mapping \(\quad \mathbf{x} \mapsto A \mathbf{x}\) d. If the equation \(A \mathbf{x}=\mathbf{b}\) is consistent, then \(\operatorname{Col} A\) is \(\mathbb{R}^{m} .\) e. The kernel of a linear transformation is a vector space. f. \(\mathrm{Col} A\) is the set of all vectors that can be written as \(A \mathbf{x}\) for some \(\mathbf{x} .\)