91Ó°ÊÓ

(a) determine a basis for rowspace \((A)\) and make a sketch of it in the \(x y\) -plane; (b) Repeat part (a) for colspace \((A)\). $$A=\left[\begin{array}{rr} 6 & -1 \\ 12 & -2 \end{array}\right]$$

Let \(S=\left\\{\mathbf{x} \in \mathbb{R}^{3}: \mathbf{x}=(r-2 s, 3 r+s, s), r, s \in \mathbb{R}\right)\) (a) Show that \(S\) is a subspace of \(\mathbb{R}^{3}\). (b) Show that the vectors in \(S\) lie on the plane with equation \(3 x-y+7 z=0\).

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$V=\mathbb{R}^{2} ; B=\\{(7,-1),(-9,-2)\\} ; \mathbf{v}=(27,6)$$

If \(\mathbf{x}=(-1,-4)\) and \(\mathbf{y}=(-5,1),\) determine the vectors \(\mathbf{v}_{1}=3 \mathbf{x}, \mathbf{v}_{2}=-4 \mathbf{y}, \mathbf{v}_{3}=3 \mathbf{x}+(-4) \mathbf{y} .\) Sketch the corresponding points in the \(x y\) -plane and the equivalent geometric vectors.

Determine the null space of \(A\) and verify the Rank-Nullity Theorem. $$A=\left[\begin{array}{rr}2 & -1 \\\\-4 & 2\end{array}\right]$$

Determine whether the given set (together with the usual operations on that set) forms a vector space over \(\mathbb{R}\). In all cases, justify your answer carefully. The set of polynomials of degree 5 or less whose coefficients are even integers.

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$\begin{array}{l} V=\mathbb{R}^{3} ; B=\\{(1,0,1),(1,1,-1),(2,0,1)\\} \\ \mathbf{v}=(-9,1,-8) \end{array}$$

Determine whether the given set \(S\) of vectors is closed under addition and closed under scalar multiplication. In each case, take the set of scalars to be the set of all real numbers. The set \(S:=\left\\{a_{0}+a_{1} x+a_{2} x^{2}: a_{0}+a_{1}+a_{2}=1\right\\}.\)

Determine whether the given set \(S\) of vectors is closed under addition and closed under scalar multiplication. In each case, take the set of scalars to be the set of all real numbers. The set \(S\) of all solutions to the differential equation \(y^{\prime}+3 y=0 .\) (Do not solve the differential equation.)

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=M_{2}(\mathrm{R}),\) and \(S\) is the subset of all \(2 \times 2\) matrices with \(\operatorname{det}(A)=1\).