Chapter 4

Show with examples that if \(x\) is a vector in the first quadrant of \(\mathbb{R}^{2}\) (i.e., both coordinates of \(\mathbf{x}\) are positive) and \(y\) is a vector in the third quadrant of \(\mathbb{R}^{2}\) (i.e., both coordinates of y are negative), then the sum \(x+y\) could occur in any of the four quadrants.

Show that a \(3 \times 7\) matrix \(A\) with nullity \((A)=4\) must have colspace \((A)=\mathbb{R}^{3} .\) Is rowspace \((A)=\mathbb{R}^{3} ?\)

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B.\) $$\begin{array}{l}V=M_{2}(\mathbb{R}); \\\B=\left\\{\left[\begin{array}{cc}-1 & 1 \\\0 & 1 \end{array}\right],\left[\begin{array}{rr}1 & 3 \\\\-1 & 0\end{array}\right],\left[\begin{array}{cc} 1 & 0 \\\1 & 2\end{array}\right],\left[\begin{array}{cc}0 & -1 \\\2 & 3\end{array}\right]\right\\} \\ A=\left[\begin{array}{cc}5 & 6 \\\7 & 8\end{array}\right]\end{array}$$

Let \(A=\left[\begin{array}{rrr}1 & 2 & 4 \\ 5 & 11 & 21 \\ 3 & 7 & 13\end{array}\right]\). (a) Find a basis for rowspace \((A)\) and colspace \((A)\) (b) Show that rowspace( \(A\) ) corresponds to the plane with Cartesian equation \(2 x+y-z=0,\) whereas colspace \((A)\) corresponds to the plane with Cartesian equation \(2 x-y+z=0\).

Determine all values of the constant \(k\) for which the given set of vectors is linearly independent in \(\mathbb{R}^{4}\). \(\\{(1,0,1, k),(-1,0, k, 1),(2,0,1,3)\\}\)

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 the form \(a+b x^{3}+c x^{4}\) where \(a, b, c \in \mathbb{R}\).

Show that \(\mathbf{v}_{1}=(-1,3,2), \mathbf{v}_{2}=(1,-2,1), \mathbf{v}_{3}=\) (2,1,1) span \(\mathbb{R}^{3},\) and express \(\mathbf{v}=(x, y, z)\) 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 \(V\). \(V=M_{2 \times 3}(\mathbb{R}),\) and \(S\) is the subset of all \(2 \times 3\) matrices such that the elements in each row sum to \(10 .\)

Show that \\{(1,2),(3,8)\\} is a linearly dependent set in the vector space \(V\) in Problem 13

Show that a \(6 \times 4\) matrix \(A\) with nullity \((A)=0\) must have rowspace \((A)=\mathbb{R}^{4} .\) Is colspace \((A)=\mathbb{R}^{4} ?\)

Determine whether the given set \(S\) of vectors is a basis for \(M_{m \times n}(\mathbb{R})\). $$ \begin{array}{l} m=n=2: S=\left\\{\left[\begin{array}{rr} -3 & 1 \\ 0 & 2 \end{array}\right],\left[\begin{array}{rr} 3 & -5 \\ 6 & 1 \end{array}\right]\right. \\ \left.\left[\begin{array}{rr} -1 & -2 \\ 1 & 0 \end{array}\right],\left[\begin{array}{rr} 0 & 3 \\ 1 & -4 \end{array}\right],\left[\begin{array}{rr} 6 & -2 \\ -3 & -4 \end{array}\right]\right\\} \end{array} $$

Let \(\mathbf{v}_{1}=(0,6,3), \mathbf{v}_{2}=(3,0,3),\) and \(\mathbf{v}_{3}=\) \((6,-3,0) .\) Determine the component vector of an arbitrary vector \(\mathbf{v}=(x, y, z)\) relative to the basis \(\left\\{\mathbf{v}_{1}, \mathbf{v}_{2}, \mathbf{v}_{3}\right\\}\).

For Problems \(14-15\), determine all values of the constant \(k\) for which the given set of vectors is linearly independent in \(\mathbb{R}^{4}\). \(\\{(1,1,0,-1),(1, k, 1,1),(2,1, k, 1),(-1,1,1, k)\\}\)

We have defined the set \(\mathrm{R}^{2}=\\{(x, y): x, y \in \mathbb{R}\\},\) together with the addition and scalar multiplication operations as follows: $$\begin{aligned} \left(x_{1}, y_{1}\right)+\left(x_{2}, y_{2}\right) &=\left(x_{1}+x_{2}, y_{1}+y_{2}\right) \\ k\left(x_{1}, y_{1}\right) &=\left(k x_{1}, k y_{1}\right) \end{aligned}$$ Give a complete verification that each of the vector space axioms is satisfied.

Show that \(\mathbf{v}_{1}=(1,1), \mathbf{v}_{2}=(-1,2), \mathbf{v}_{3}=(1,4)\) span \(\mathbb{R}^{2} .\) Do \(\mathbf{v}_{1}, \mathbf{v}_{2}\) alone span \(\mathbb{R}^{2}\) also?

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 \(2 \times 2\) real symmetric matrices.

For Problems \(15-18\), determine whether the given set \(S\) of vectors is a basis for \(M_{m \times n}(\mathbb{R})\). $$ \begin{array}{l} m=n=2: S=\left\\{\left[\begin{array}{rr} -2 & -8 \\ 1 & 4 \end{array}\right],\left[\begin{array}{rr} 0 & 1 \\ -1 & 1 \end{array}\right],\right. \\ \left.\left[\begin{array}{rr} -5 & 0 \\ 5 & -4 \end{array}\right],\left[\begin{array}{ll} 3 & -2 \\ 4 & -1 \end{array}\right]\right\\} \end{array} $$

Let \(p_{1}(x)=1+x, p_{2}(x)=-x+x^{2},\) and \(p_{3}(x)=\) \(1+2 x^{2} .\) Determine the component vector of an arbitrary polynomial \(p(x)=a_{0}+a_{1} x+a_{2} x^{2}\) relative to the basis \(\left\\{p_{1}, p_{2}, p_{3}\right\\}.\)

Give examples to show how each type of elementary row operation applied to a matrix can change the column space of the matrix.

Determine whether the given set of vectors is linearly independent in \(M_{2}(\mathbb{R})\). $$A_{1}=\left[\begin{array}{ll} 1 & 1 \\ 0 & 1 \end{array}\right], A_{2}=\left[\begin{array}{rr} 2 & -1 \\ 0 & 1 \end{array}\right], A_{3}=\left[\begin{array}{ll} 3 & 6 \\ 0 & 4 \end{array}\right]$$.

Determine the zero vector in the vector space \(V=\) \(M_{4 \times 2}(\mathrm{R}),\) and write down a general element \(A\) in \(V\) along with its additive inverse \(-A\).

Let \(S\) be the subspace of \(\mathbb{R}^{3}\) consisting of all vectors of the form \(\mathbf{v}=\left(c_{1}, c_{2}, c_{2}-2 c_{1}\right) .\) Determine a set of vectors that spans \(S .\)

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V\) is the vector space of all real-valued functions defined on the interval \([a, b],\) and \(S\) is the subset of \(V\) consisting of all real-valued functions \([a, b]\) satisfying \(f(a)=5 \cdot f(b)\)

What is the dimension of the subspace of \(P_{2}(\mathbb{R})\) given by $$ S=\operatorname{span}\left\\{2+x^{2}, 4-2 x+3 x^{2}, 1+x\right\\} ? $$

Show that a \(5 \times 7\) matrix \(A\) must have \(2 \leq\) nullity \((A) \leq 7 .\) Give an example of a \(5 \times 7\) matrix \(A\) with nullity \((A)=2\) and a \(5 \times 7\) matrix \(A\) with nullity \((A)=7\).

Determine whether the given set \(S\) of vectors is a basis for \(M_{m \times n}(\mathbb{R})\). \(m=3, n=2: S=\left\\{\left[\begin{array}{rr}6 & -3 \\ 1 & 4 \\ 4 & -4\end{array}\right],\left[\begin{array}{rr}0 & -2 \\ 9 & 1 \\ -3 & -5\end{array}\right]\right.\) \(\left.\left[\begin{array}{rr}2 & -9 \\ 1 & 1 \\\ -3 & 0\end{array}\right],\left[\begin{array}{rr}1 & -5 \\ 2 & 0 \\ -4 & 0\end{array}\right],\left[\begin{array}{rr}-7 & 5 \\ 0 & -1 \\ 3 & 1\end{array}\right]\right\\}\)

Find the change-of-basis matrix \(P_{C \leftarrow B}\) from the given ordered basis \(B\) to the given ordered basis \(C\) of the vector space \(V.\) $$\begin{aligned}&V=\mathbb{R}^{2} ; B=\\{(9,2),(4,-3)\\}\\\&C=\\{(2,1),(-3,1)\\}\end{aligned}$$

Let \(A\) be an \(m \times n\) matrix with colspace \((A)=\) nullspace(A). Prove that \(m=n\).

Determine whether the given set of vectors is linearly independent in \(M_{2}(\mathbb{R})\). $$A_{1}=\left[\begin{array}{rr} 2 & -1 \\ 3 & 4 \end{array}\right], A_{2}=\left[\begin{array}{rr} -1 & 2 \\ 1 & 3 \end{array}\right]$$.

Let \(S\) be the subspace of \(\mathbb{R}^{4}\) consisting of all vectors of the form \(\mathbf{v}=\left(c_{1}, c_{2}, c_{2}-c_{1}, c_{1}-2 c_{2}\right) .\) Determine a set of vectors that spans \(S .\)

Determine the solution set to \(A \mathbf{x}=\mathbf{b}\) and show that all solutions are of the form (4.9.3). Show that \(3 \times 8\) matrix \(A\) must have \(5 \leq\) nullity \((A) \leq\) 8. Give an example of a \(3 \times 8\) matrix \(A\) with nullity \((A)=5\) and a \(3 \times 8\) matrix \(A\) with nullity \((A)=8\).

Find the change-of-basis matrix \(P_{C \leftarrow B}\) from the given ordered basis \(B\) to the given ordered basis \(C\) of the vector space \(V.\) $$\begin{aligned}&V=\mathbb{R}^{2} ; B=\\{(-5,-3),(4,28)\\}\\\&C=\\{(6,2),(1,-1)\\}\end{aligned}$$

Let \(A\) be an \(n \times n\) matrix with rowspace( \(A\) ) \(=\) nullspace( \(A\) ). Prove that \(A\) cannot be invertible.

Determine whether the given set of vectors is linearly independent in \(M_{2}(\mathbb{R})\). $$A_{1}=\left[\begin{array}{ll} 1 & 0 \\ 1 & 2 \end{array}\right], A_{2}=\left[\begin{array}{rr} -1 & 1 \\ 2 & 1 \end{array}\right], A_{3}=\left[\begin{array}{ll} 2 & 1 \\ 5 & 7 \end{array}\right]$$.

Determine the zero vector in the vector space \(V=\) \(P_{3}(\mathbb{R}),\) and write down a general element \(p(x)\) in \(V\) along with its additive inverse \(-p(x)\).

Let \(S\) be the subspace of \(M_{2}(\mathbb{R})\) consisting of all skewsymmetric \(2 \times 2\) matrices with real elements. Determine a matrix that spans \(S .\)

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V\) is the vector space of all real-valued functions defined on the interval \((-\infty, \infty),\) and \(S\) is the subset of \(V\) consisting of all real- valued functions satisfying \(f(-x)=f(x)\) for all \(x \in(-\infty, \infty)\).

Decide (with justification) whether \(S\) is a subspace of \(V\) $$V=\mathbb{R}^{2}, S=\left\\{\left(x, x^{3}\right): x \in \mathbb{R}\right\\}$$

Prove that if \(A\) and \(B\) are \(n \times n\) matrices and \(A\) is invertible, then nullity \((A B)=\) nullity \((B)=\) nullity \((B A)\) [Hint: \(B \mathbf{x}=\mathbf{0} \text { if and only if } A B \mathbf{x}=\mathbf{0 .}]\)

Find the change-of-basis matrix \(P_{C \leftarrow B}\) from the given ordered basis \(B\) to the given ordered basis \(C\) of the vector space \(V.\) $$\begin{array}{l}V=\mathbb{R}^{3} ; B=\\{(2,-5,0),(3,0,5),(8,-2,-9)\\} \\\C=\\{(1,-1,1),(2,0,1),(0,1,3)\\} \end{array}.$$

determine whether the given set of vectors is linearly independent in \(P_{2}(\mathbb{R})\). $$p_{1}(x)=1-x, \quad p_{2}(x)=1+x$$.

Let \(S\) be the subset of \(M_{2}(\mathbb{R})\) consisting of all upper triangular \(2 \times 2\) matrices. (a) Verify that \(S\) is a subspace of \(M_{2}(\mathbb{R})\) (b) Determine a set of \(2 \times 2\) matrices that span \(S .\)

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=P_{2}(\mathbb{R}),\) and \(S\) is the subset of \(P_{2}(\mathbb{R})\) consisting of all polynomials of the form \(p(x)=a x^{2}+b\).

Decide (with justification) whether \(S\) is a subspace of \(V\) $$V=M_{2}(\mathbb{R}), S=\\{2 \times 2 \text { orthogonal matrices }\\}$$

Find the change-of-basis matrix \(P_{C \leftarrow B}\) from the given ordered basis \(B\) to the given ordered basis \(C\) of the vector space \(V.\) $$\begin{array}{l}V=\mathbb{R}^{3} ; B=\\{(-7,4,4),(4,2,-1),(-7,5,0)\\} \\\C=\\{(1,1,0),(0,1,1),(3,-1,-1)\\} \end{array}.$$

determine whether the given set of vectors is linearly independent in \(P_{2}(\mathbb{R})\). $$p_{1}(x)=2+3 x, \quad p_{2}(x)=4+6 x$$.

On \(\mathbb{R}^{+}\), the set of positive real numbers, define the operations of addition, \(\oplus,\) and scalar multiplication, \(\odot\) as follows: $$\begin{array}{l} x \oplus y=x y \\ c \odot x=x^{c}. \end{array}$$ Note that the multiplication and exponentiation appearing on the right side of these formulas refer to the ordinary operations on real numbers. Determine whether \(\mathbb{R}^{+}\), together with these algebraic operations, is a vector space.

Let \(S\) be the subspace of \(M_{2}(\mathbb{R})\) consisting of all \(2 \times 2\) matrices whose four elements sum to zero (see Problem 12 in Section 4.3 ). Find a set of vectors that spans \(S\)

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=P_{2}(\mathbb{R}),\) and \(S\) is the subset of \(P_{2}(\mathbb{R})\) consisting of all polynomials of the form \(p(x)=a x^{2}+1\).

Decide (with justification) whether \(S\) is a subspace of \(V\) $$V=C[a, b], S=\\{f \in V: f(a)=2 f(b)\\}$$