Chapter 4

Let \(r\) and \(s\) denote scalars and let \(\mathbf{v}\) and \(\mathbf{w}\) denote vectors in \(\mathbb{R}^{5}\). $$\text { Prove that }(r+s) \mathbf{v}=r \mathbf{v}+s \mathbf{v}$$

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

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

determine whether the given set of vectors is linearly independent or linearly dependent in \(\mathbb{R}^{n} .\) In the case of linear dependence, find a dependency relationship. $$\\{(3,6,9)\\}$$.

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 \(S:=\mathbb{Q}\) of all rational numbers.\(^{1}\)

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 \(3 x-y+7 z=0\).

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.

Let \(r\) and \(s\) denote scalars and let \(\mathbf{v}\) and \(\mathbf{w}\) denote vectors in \(\mathbb{R}^{5}\). $$\text { Prove that } r(\mathbf{v}+\mathbf{w})=r \mathbf{v}+r \mathbf{w}$$

Determine the null space of \(A\) and verify the Rank-Nullity Theorem. $$A=\left[\begin{array}{llll}1 & 0 & -6 & -1\end{array}\right]$$

(a) determine a basis for rowspace \((A)\) and make a sketch of it in the \(x y\) -plane; (b) Repeat part (a) for colspace \((A)\). $$A=\left[\begin{array}{rr} 1 & -2 \\ -3 & 6 \end{array}\right]$$

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$V=\mathbb{R}^{2} ; B=\\{(2,-2),(1,4)\\} ; \mathbf{v}=(5,-10)$$

Determine whether the given set of vectors spans \(\mathbb{R}^{2}\). $$\\{(1,-1),(2,-2),(2,3)\\}$$

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 \(S:=U_{n}(\mathbb{R})\) of all upper triangular \(n \times n\) matrices with real elements.

Let \(S=\left\\{\mathbf{x} \in \mathbb{R}^{2}: \mathbf{x}=(2 k,-3 k), k \in \mathbb{R}\right\\}\) (a) Show that \(S\) is a subspace of \(\mathbb{R}^{2}\). (b) Make a sketch depicting the subspace \(S\) in the Cartesian plane.

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

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. $$V=\left\\{p(x) \in P_{2}(\mathbb{R}): p(3)=0 \text { and } p^{\prime}(5)=0\right\\}$$

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

(a) find \(n\) such that rowspace \((A)\) is a subspace of \(\mathbb{R}^{n}\), and determine a basis for rowspace \((A) ;\) (b) find \(m\) such that colspace \((A)\) is a subspace of \(\mathbb{R}^{m},\) and determine a basis for colspace \((A)\). $$A=\left[\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \end{array}\right]$$

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

determine whether the given set of vectors is linearly independent or linearly dependent in \(\mathbb{R}^{n} .\) In the case of linear dependence, find a dependency relationship. $$\\{(2,-1),(3,2),(0,1)\\}$$.

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=\mathbb{R}^{3},\) and \(S\) is the set of all vectors \((x, y, z)\) in \(V\) such that \(z=3 x\) and \(y=2 x\).

If \(\mathbf{x}=(5,-2,9)\) and \(\mathbf{y}=(-1,6,4),\) determine the additive inverse of the vector \(\mathbf{v}=-2 \mathbf{x}+10 \mathbf{y}\)

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 null space of \(A\) and verify the Rank-Nullity Theorem. $$A=\left[\begin{array}{rrr}1 & 1 & -1 \\\3 & 4 & 4 \\\1 & 1 & 0\end{array}\right]$$

(a) find \(n\) such that rowspace \((A)\) is a subspace of \(\mathbb{R}^{n}\), and determine a basis for rowspace \((A) ;\) (b) find \(m\) such that colspace \((A)\) is a subspace of \(\mathbb{R}^{m},\) and determine a basis for colspace \((A)\). $$A=\left[\begin{array}{r} -3 \\ 1 \\ 7 \end{array}\right]$$

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 of vectors is linearly independent or linearly dependent in \(\mathbb{R}^{n} .\) In the case of linear dependence, find a dependency relationship. $$\\{(1,-1,0),(0,1,-1),(1,1,1)\\}$$.

Determine whether the given set of vectors spans \(\mathbb{R}^{2}\). $$\\{(6,-2),(-2,2 / 3),(3,-1)\\}$$

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 \(S:=\left\\{a_{0}+a_{1} x+a_{2} x^{2}: a_{0}+a_{1}+a_{2}=0\right\\}.\)

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=\mathbb{R}^{2},\) and \(S\) is the set of all vectors \((x, y)\) in \(V\) satisfying \(3 x+2 y=0\).

If \(\mathbf{x}=(3,-1,2,5)\) and \(\mathbf{y}=(-1,2,9,-2),\) determine \(\mathbf{v}=5 \mathbf{x}+(-7) \mathbf{y}\) and its additive inverse.

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 all polynomials of degree 5 or less whose coefficients of \(x^{2}\) and \(x^{3}\) are zero.

Determine the null space of \(A\) and verify the Rank-Nullity Theorem. $$A=\left[\begin{array}{llll}1 & 4 & -1 & 3 \\\2 & 9 & -1 & 7 \\\2 & 8 & -2 & 6\end{array}\right]$$

(a) find \(n\) such that rowspace \((A)\) is a subspace of \(\mathbb{R}^{n}\), and determine a basis for rowspace \((A) ;\) (b) find \(m\) such that colspace \((A)\) is a subspace of \(\mathbb{R}^{m},\) and determine a basis for colspace \((A)\). $$A=\left[\begin{array}{llll} 1 & 1 & -3 & 2 \\ 3 & 4 & -11 & 7 \end{array}\right]$$

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$\begin{array}{l} V=\mathbb{R}^{3} ; B=\\{(1,-6,3),(0,5,-1),(3,-1,-1)\\} \\ \mathbf{v}=(1,7,7) \end{array}$$

determine whether the given set of vectors is linearly independent or linearly dependent in \(\mathbb{R}^{n} .\) In the case of linear dependence, find a dependency relationship. $$\\{(1,2,3),(1,-1,2),(1,-4,1)\\}$$.

Recall that three vectors \(\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\) in \(\mathbb{R}^{3}\) are coplanar if and only if $$ \operatorname{det}\left(\left[\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right]\right)=0 $$ Use this result to determine whether the given set of vectors spans \(\mathbb{R}^{3}\). $$\\{(1,-1,1),(2,5,3),(4,-2,1)\\}$$

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 \(S:=\left\\{a_{0}+a_{1} x+a_{2} x^{2}: a_{0}+a_{1}+a_{2}=1\right\\}.\)

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=\mathbb{R}^{4},\) and \(S\) is the set of all vectors of the form \(\left(x_{1}, 0, x_{3}, 2\right)\).

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 solutions to the linear system $$ \begin{aligned} -2 x_{2}+5 x_{3} &=7 \\ 4 x_{1}-6 x_{2}+3 x_{3} &=0 \end{aligned} $$

Determine the nullity of \(A\) "by inspection" by appealing to the Rank-Nullity Theorem. Avoid computations. $$A=\left[\begin{array}{rr}2 & -3 \\\0 & 0 \\\\-4 & 6 \\\22 & -33\end{array}\right]$$

(a) find \(n\) such that rowspace \((A)\) is a subspace of \(\mathbb{R}^{n}\), and determine a basis for rowspace \((A) ;\) (b) find \(m\) such that colspace \((A)\) is a subspace of \(\mathbb{R}^{m},\) and determine a basis for colspace \((A)\). $$A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 5 & 6 & 7 \\ 9 & 10 & 11 \end{array}\right]$$

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$\begin{array}{l} V=\mathbb{R}^{3} ; B=\\{(3,-1,-1),(1,-6,3),(0,5,-1)\\} \\ \mathbf{v}=(1,7,7) \end{array}$$

determine whether the given set of vectors is linearly independent or linearly dependent in \(\mathbb{R}^{n} .\) In the case of linear dependence, find a dependency relationship. $$\\{(-2,4,-6),(3,-6,9)\\}$$.

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=\mathbb{R}^{3},\) and \(S\) is the set of all vectors \((x, y, z)\) in satisfying \(x+y+z=1\).

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 solutions to the linear system $$ \begin{array}{l} 4 x_{1}-7 x_{2}+2 x_{3}=0 \\ 5 x_{1}-2 x_{2}+9 x_{3}=0 \end{array} $$

Determine the nullity of \(A\) "by inspection" by appealing to the Rank-Nullity Theorem. Avoid computations. $$A=\left[\begin{array}{rrrrr}1 & 3 & -3 & 2 & 5 \\\\-4 & -12 & 12 & -8 & -20 \\\0 & 0 & 0 & 0 & 0 \\\1& 3 & -3 & 2 & 6\end{array}\right]$$

(a) find \(n\) such that rowspace \((A)\) is a subspace of \(\mathbb{R}^{n}\), and determine a basis for rowspace \((A) ;\) (b) find \(m\) such that colspace \((A)\) is a subspace of \(\mathbb{R}^{m},\) and determine a basis for colspace \((A)\). $$A=\left[\begin{array}{rrr} 0 & 3 & 1 \\ 0 & -6 & -2 \\ 0 & 12 & 4 \end{array}\right]$$

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$\begin{aligned} &V=\mathbb{R}^{3} ; B=\\{(-1,0,0),(0,0,-3),(0,-2,0)\\}\\\ &\mathbf{v}=(5,5,5) \end{aligned}$$

determine whether the given set of vectors is linearly independent or linearly dependent in \(\mathbb{R}^{n} .\) In the case of linear dependence, find a dependency relationship. $$\\{(1,-1,2),(2,1,0)\\}$$.