Chapter 4

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.)

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 \(\mathbb{R}^{3}\). $$\\{(2,-1,4),(3,-3,5),(1,1,3)\\}$$

If \(\mathbf{x}=(-2+i, 3 i)\) and \(\mathbf{y}=(5,2-2 i)\) in \(\mathbb{C}^{2},\) find a vector \(\mathbf{z}\) in \(\mathbb{C}^{2}\) such that \((1+i) \mathbf{x}-2 \mathbf{y}=2 i \mathbf{z}\)

Determine the nullity of \(A\) "by inspection" by appealing to the Rank-Nullity Theorem. Avoid computations. $$A=\left[\begin{array}{lll}0 & 1 & 0 \\\0 & 1 & 0 \\\0 & 0 & 1 \\\0 & 0 & 1\end{array}\right]$$

(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 \((A)\). $$A=\left[\begin{array}{rrrr} 1 & 2 & -1 & 3 \\ 3 & 6 & -3 & 5 \\ 1 & 2 & -1 & -1 \\ 5 & 10 & -5 & 7 \end{array}\right]$$

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. $$\\{(-1,1,2),(0,2,-1),(3,1,2),(-1,-1,1)\\}$$.

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$\begin{aligned} &V=P_{2}(\mathbb{R}) ; B=\left\\{x^{2}+x, 2+2 x, 1\right\\}\\\ &p(x)=-4 x^{2}+2 x+6 \end{aligned}$$

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. For a fixed \(m \times n\) matrix \(A,\) the set $$S:=\left\\{\mathbf{x} \in \mathbb{R}^{n}: A \mathbf{x}=\mathbf{0}\right\\}.$$ (This is the set of all solutions to the homogeneous linear system of equations \(A \mathbf{x}=\mathbf{0}\) and is often called the null space of \(A .\) )

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 \(\mathbb{R}^{3}\). $$\\{(1,2,3),(4,5,6),(7,8,9)\\}$$

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

If \(\mathbf{x}=(5+i, 0,-1-2 i, 1+8 i)\) and \(\mathbf{y}=(-3, i, i, 3)\) in \(\mathrm{C}^{4},\) find a vector \(\mathbf{z}\) in \(\mathrm{C}^{4}\) such that \(2 \mathbf{x}+3 i \mathbf{y}=(1+i) \mathbf{z}\)

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 \(2 \times 2\) real matrices that commute with the matrix \(\left[\begin{array}{ll}1 & 2 \\ 0 & 2\end{array}\right]\)

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

(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 \((A)\). $$A=\left[\begin{array}{rrrr} 1 & -1 & 2 & 3 \\ 1 & 1 & -2 & 6 \\ 3 & 1 & 4 & 2 \end{array}\right]$$

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$\begin{aligned} &V=P_{2}(\mathbb{R}) ; B=\left\\{5-3 x, 1,1+2 x^{2}\right\\}\\\ &p(x)=15-18 x-30 x^{2} \end{aligned}$$

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. $$\\{(1,-1,2,3),(2,-1,1,-1),(-1,1,1,1)\\}$$.

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 \(S:=\left\\{A \in M_{2}(\mathbb{R}): \operatorname{det}(A)=0\right\\}.\)

Show that the set of vectors $$ \\{(-4,1,3),(5,1,6),(6,0,2)\\} $$ does not span \(\mathbb{R}^{3},\) but that it does span the subspace of \(\mathbb{R}^{3}\) consisting of all vectors lying in the plane with equation \(x+13 y-3 z=0\)

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 \(\operatorname{det}(A)=1\).

Verify the commutative law of addition for vectors in \(\mathbb{R}^{4}\).

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 functions \(f:[0,1] \rightarrow[0,1]\) such that \(f(0)=f\left(\frac{1}{4}\right)=f\left(\frac{1}{2}\right)=f\left(\frac{3}{4}\right)=f(1)=0\)

Determine the solution set to \(A \mathbf{x}=\mathbf{b}\) and show that all solutions are of the form (4.9.3). $$A=\left[\begin{array}{rrr}1 & 3 & -1 \\\2 & 7 & 9 \\\1 & 5 & 21\end{array}\right], \mathbf{b}=\left[\begin{array}{r}4 \\\11 \\\10\end{array}\right]$$

Determine a basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors by (a) using the concept of the row space of a matrix, and (b) using the concept of the column space of a matrix. $$\\{(1,-1,2),(5,-4,1),(7,-5,-4)\\}$$

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$\begin{array}{l} V=P_{3}(\mathbb{R}) ; B=\left\\{1,1+x, 1+x+x^{2}, 1+x+x^{2}+x^{3}\right\\} \\ p(x)=4-x+x^{2}-2 x^{3} \end{array}$$

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. $$\begin{aligned} &\\{(2,-1,0,1),(1,0,-1,2),(0,3,1,2)\\\ &(-1,1,2,1)\\} \end{aligned}$$.

Show that the set of vectors $$ \\{(1,2,3),(3,4,5),(4,5,6)\\} $$ does not span \(\mathbb{R}^{3},\) but that it does span the subspace of \(\mathbb{R}^{3}\) consisting of all vectors lying in the plane with equation \(x-2 y+z=0\)

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=M_{n}(\mathbb{R}),\) and \(S\) is the subset of all \(n \times n\) lower triangular matrices.

Verify the associative law of addition for vectors in \(\mathbb{R}^{4}\).

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 functions \(f:[0,1] \rightarrow[0,1]\) such that \(|f(x)| \leq x\) for all \(x\) in [0,1]

Determine the solution set to \(A \mathbf{x}=\mathbf{b}\) and show that all solutions are of the form (4.9.3). $$A=\left[\begin{array}{llll}2 & -1 & 1 & 4 \\\1 & -1 & 2 & 3 \\\1 & -2 & 5 & 5\end{array}\right], \mathbf{b}=\left[\begin{array}{r}5 \\\6 \\\13\end{array}\right]$$

Determine a basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors by (a) using the concept of the row space of a matrix, and (b) using the concept of the column space of a matrix. $$\\{(1,3,3),(1,5,-1),(2,7,4),(1,4,1)\\}$$

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$\begin{array}{l} V=P_{3}(\mathbb{R}) ; B=\left\\{x^{3}+x^{2}, x^{3}-1, x^{3}+1, x^{3}+x\right\\} \\ p(x)=8+x+6 x^{2}+9 x^{3} \end{array}$$

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 \(S:=\left\\{(x, y) \in \mathbb{R}^{2}: y=x+1\right\\}.\)

Show that \(\mathbf{v}_{1}=(2,-1), \mathbf{v}_{2}=(3,2)\) span \(\mathbb{R}^{2}\) and express the vector \(\mathbf{v}=(5,-7)\) as a linear combination of \(\mathbf{v}_{1}, \mathbf{v}_{2}\)

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=M_{n}(\mathbb{R}),\) and \(S\) is the subset of all \(n \times n\) invertible matrices.

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 \(n \times n\) matrices \(A\) such that \(A^{2}\) is symmetric.

Determine the solution set to \(A \mathbf{x}=\mathbf{b}\) and show that all solutions are of the form (4.9.3). $$A=\left[\begin{array}{rrr}1 & 1 & -2 \\\3 & -1 & -7 \\\1 & 1 & 1 \\\2 & 2 & -4\end{array}\right], \mathbf{b}=\left[\begin{array}{r}-3 \\\2 \\\0 \\\\-6\end{array}\right]$$

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$\begin{aligned} &V=M_{2}(\mathbb{R})\\\ &B=\left\\{\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right],\left[\begin{array}{ll} 1 & 1 \\ 1 & 0 \end{array}\right],\left[\begin{array}{ll} 1 & 1 \\ 0 & 0 \end{array}\right],\left[\begin{array}{ll} 1 & 0 \\ 0 & 0 \end{array}\right]\right\\} \end{aligned}$$ $$A=\left[\begin{array}{cc} -3 & -2 \\ -1 & 2 \end{array}\right]$$

Determine a basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors by (a) using the concept of the row space of a matrix, and (b) using the concept of the column space of a matrix. $$\\{(1,1,-1,2),(2,1,3,-4),(1,2,-6,10)\\}$$

Consider the vectors \(\mathbf{v}_{1}=(2,-1,5), \mathbf{v}_{2}=(1,3,-4)\) \(\mathbf{v}_{3}=(-3,-9,12)\) in \(\mathbb{R}^{3}\) (a) Show that \(\left\\{v_{1}, v_{2}, v_{3}\right\\}\) is linearly dependent. (b) Is \(\mathbf{v}_{1} \in \operatorname{span}\left\\{\mathbf{v}_{2}, \mathbf{v}_{3}\right\\} ?\) Draw a picture illustrating your answer.

Show that \(\mathbf{v}_{1}=(1,-5), \mathbf{v}_{2}=(6,3)\) span \(\mathbb{R}^{2},\) and express the vector \(\mathbf{v}=(x, y)\) as a linear combination of \(\mathbf{v}_{1}, \mathbf{v}_{2}\)

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=M_{2}(\mathbb{R}),\) and \(S\) is the subset of all \(2 \times 2\) matrices whose four elements sum to zero.

Let $$ V=\left\\{\left(a_{1}, a_{2}\right): a_{1}, a_{2} \in \mathbb{R}, a_{2}>0\right\\} $$ Define addition and scalar multiplication on \(V\) as follows: $$\begin{aligned} \left(a_{1}, a_{2}\right) & \oplus\left(b_{1}, b_{2}\right)=\left(a_{1}+b_{1}, a_{2} b_{2}\right) \\ k & \otimes\left(a_{1}, a_{2}\right)=\left(k a_{1}, a_{2}^{k}\right), \quad k \in \mathbb{R} \end{aligned}$$ Explicitly verify that \(V\) is a vector space over \(\mathbb{R}\).

Determine the solution set to \(A \mathbf{x}=\mathbf{b}\) and show that all solutions are of the form (4.9.3). $$A=\left[\begin{array}{llll}1 & 1 & -1 & 5 \\\0 & 2 & -1 & 7 \\\4 & 2 & -3 & 13\end{array}\right], \mathbf{b}=\left[\begin{array}{l}0 \\\0 \\\0\end{array}\right]$$

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$\begin{aligned} &V=M_{2}(\mathbb{R})\\\ &B=\left\\{\left[\begin{array}{cc} 2 & -1 \\ 3 & 5 \end{array}\right],\left[\begin{array}{rr} 0 & 4 \\ -1 & 1 \end{array}\right],\left[\begin{array}{ll} 1 & 1 \\ 1 & 1 \end{array}\right],\left[\begin{array}{rr} 3 & -1 \\ 2 & 5 \end{array}\right]\right\\}\\\ &A=\left[\begin{array}{rr} -10 & 16 \\ -15 & -14 \end{array}\right] \end{aligned}$$

Determine a basis for the subspace of \(\mathbb{R}^{n}\) spanned by the given set of vectors by (a) using the concept of the row space of a matrix, and (b) using the concept of the column space of a matrix. $$\\{(1,4,1,3),(2,8,3,5),(1,4,0,4),(2,8,2,6)\\}$$

Determine all values of the constant \(k\) for which the vectors \((1,1, k),(0,2, k)\) and \((1, k, 6)\) are linearly dependent in \(\mathbb{R}^{3}\).

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 \(S\) of all polynomials of degree exactly 2.

Show that \(\mathbf{v}_{1}=(1,-3,2), \mathbf{v}_{2}=(1,0,-1), \mathbf{v}_{3}=\) (1,2,-4) span \(\mathbb{R}^{3},\) and express \(\mathbf{v}=(9,8,7)\) as a linear combination of \(\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\)

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=M_{3 \times 2}(\mathbb{R}),\) and \(S\) is the subset of all \(3 \times 2\) matrices such that the elements in each column sum to zero.