Determine whether the following sets form subspaces of ${ℝ}^2$:

1. $\{(x_1,x_2{)}^T|x_1+x_2=0\}$

2. $\{(x_1,x_2{)}^T|x_1{x}_2=0\}$

3. $\{(x_1,x_2{)}^T|x_1=3x_2\}$

4. $\{(x_1,x_2{)}^T|\left|x_1\right|=\left|x_2\right|\}$

5. $\{(x_1,x_2{)}^T|x_1^2=x_2^2\}$

Determine whether the following sets form subspaces of ${ℝ}^3$:

1. $\{(x_1,x_2,x_3{)}^T|x_1+x_3=1\}$

2. $\{(x_1,x_2,x_3{)}^T|x_1=x_2=x_3\}$

3. $\{(x_1,x_2,x_3{)}^T|x_3=x_1+x_2\}$

4. $\{(x_1,x_2,x_3{)}^T|x_3=x_1\text{ or }{x}_3=x_2\}$

Determine whether the following are subspaces of ${ℝ}^{2\times 2}$:

1. The set of all $2\times 2$ diagonal matrices

2. The set of all $2\times 2$ triangular matrices

3. The set of all $2\times 2$ lower triangular matrices

4. The set of all $2\times 2$ matrices A such that $a_{12}=1$

5. The set of all $2\times 2$ matrices B such that $b_{11}=0$

6. The set of all symmetric $2\times 2$ matrices

7. The set of all singular $2\times 2$ matrices

Determine the null space of each of the following matrices:

1. $\left[\begin{array}{cc}2& 1\\ 3& 2\end{array}\right]$

2. $\left[\begin{array}{cccc}\hfill 1& \hfill 2& \hfill -3& \hfill -1\\ \hfill -2& \hfill -4& \hfill 6& \hfill 3\end{array}\right]$

3. $\left[\begin{array}{ccc}\hfill 1& \hfill 3& \hfill -4\\ \hfill 2& \hfill -1& \hfill -1\\ \hfill -1& \hfill -3& \hfill 4\end{array}\right]$

4. $\left[\begin{array}{cccc}\hfill 1& \hfill 1& \hfill -1& \hfill 2\\ \hfill 2& \hfill 2& \hfill -3& \hfill 1\\ \hfill -1& \hfill -1& \hfill 0& \hfill -5\end{array}\right]$

Determine whether the following are subspaces of $P_4$ (be careful!):

1. The set of polynomials in $P_4$ of even degree

2. The set of all polynomials of degree 3

3. The set of all polynomials $p(x)$ in $P_4$ such that $p(0)=0$

4. The set of all polynomials in $P_4$ having at least one real root

Determine whether the following are subspaces of $C[-1,1]$:

1. The set of functions f in $C[-1,1]$ such that $f(-1)=f(1)$

2. The set of odd functions in $C[-1,1]$

3. The set of continuous nondecreasing functions on $[-1,1]$

4. The set of functions f in $C[-1,1]$ such that $f(-1)=0$ and $f(1)=0$

5. The set of functions f in $C[-1,1]$ such that $f(1)=0$ or $f(1)=0$

Show that $C^n[a,b]$ is a subspace of $C[a,b]$.

Let A beafixedvectorin ${ℝ}^{n\times n}$ and let S be the set of all matrices that commute with A, that is,

Show that S is a subspace of ${ℝ}^{n\times n}$.

In each of the following, determine the subspace of ${ℝ}^{2\times 2}$ consisting of all matrices that commute with the given matrix:

1. $\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]$

2. $\left[\begin{array}{cc}0& 0\\ 1& 0\end{array}\right]$

3. $\left[\begin{array}{cc}1& 1\\ 0& 1\end{array}\right]$

4. $\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$

Let A be a particular vector in ${ℝ}^{2\times 2}$. Determine whether the following are subspaces of ${ℝ}^{2\times 2}$:

1. $S_1=\{B\in {\mathbb{R}}^{2\times 2}|\mathit{\text{BA}}=O\}$

2. $S_2=\{B\in {\mathbb{R}}^{2\times 2}|\mathit{\text{AB}}\ne \mathit{\text{BA}}\}$

3. $S_3=\{B\in {ℝ}^{2\times 2}|AB+B=O\}$

Determine whether the following are spanning sets for ${ℝ}^2$:

1. $\left\{\right[\begin{array}{c}2\\ 1\end{array}],[\begin{array}{c}3\\ 2\end{array}\left]\right\}$

2. $\left\{\right[\begin{array}{c}2\\ 3\end{array}],[\begin{array}{c}4\\ 6\end{array}\left]\right\}$

3. $\left\{\right[\begin{array}{c}-2\\ 1\end{array}],[\begin{array}{c}1\\ 3\end{array}],[\begin{array}{c}2\\ 4\end{array}\left]\right\}$

4. $\left\{\right[\begin{array}{c}-1\\ 2\end{array}],[\begin{array}{c}1\\ -2\end{array}],[\begin{array}{c}2\\ -4\end{array}\left]\right\}$

5. $\left\{\left[\begin{array}{c}1\\ 2\end{array}\right],\left[\begin{array}{r}-1\\ 1\end{array}\right]\right\}$

Which of the sets that follow are spanning sets for ${ℝ}^3$? Justify your answers.

1. $\{{(1,0,0)}^T,{(0,1,1)}^T,{(1,0,1)}^T\}$

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

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

4. $\{{(2,1,-2)}^T,{(-2,-1,2)}^T,{(4,2,-4)}^T\}$

5. $\{{(1,1,3)}^T,{(0,2,1)}^T\}$

Given

1. Is $\mathbf{x}\in \text{Span}({\mathbf{x}}_1,{\mathbf{x}}_2)$?

2. Is $\mathbf{y}\in \text{Span}({\mathbf{x}}_1,{\mathbf{x}}_2)$?

Prove your answers.

Let A be a $4\times 3$ matrix and let $\mathbf{b}\in {\mathbb{R}}^4$. How many possible solutions could the system $A\mathbf{x}=\mathbf{b}$ have if $N(A)=\{\mathbf{0}\}$? Answer the same question in the case $N(A)\ne \{\mathbf{0}\}$. Explain your answers.

Let A be a $4\times 3$ matrix and let

1. If $N(A)=\{\mathbf{0}\}$, what can you conclude about the solutions to the linear system $A\mathbf{x}=\mathbf{c}$?

2. If $N(A)\ne \{\mathbf{0}\}$, how many solutions will the system $A\mathbf{x}=\mathbf{c}$ have? Explain.

Let ${\mathbf{x}}_1$ be a particular solution to a system $A\mathbf{x}=\mathbf{b}$ and let $\{{\mathbf{z}}_1,{\mathbf{z}}_2,{\mathbf{z}}_3\}$ be a spanning set for $N(A)$. If

show that y will be a solution to $A\mathbf{x}=\mathbf{b}$ if and only if $\mathbf{y}={\mathbf{x}}_1+Z\mathbf{c}$ for some $\mathbf{c}\in {\mathbb{R}}^3$.

Figure 3.2.6 gives a geometric illustration of the solution set S to a system $A\mathbf{x}=\mathbf{b}$, where A is an $m\times 3$ matrix, $N(A)=\text{Span}({\mathbf{z}}_1,{\mathbf{z}}_2)$, and $\mathbf{b}=A{\mathbf{x}}_0$, for some ${\mathbf{x}}_0\notin N(A)$. Suppose we change b by setting it equal to $A{\mathbf{x}}_1$, where ${\mathbf{x}}_1$ is a different vector that is also not in $N(A)$. Explain the effect that this change will have on the original figure. Geometrically, how will the new solution set $S_1$ compare to the original solution set S and to $N(A)$?

Let $\{{\mathbf{x}}_1,{\mathbf{x}}_2,\dots ,{\mathbf{x}}_k\}$ be a spanning set for a vector space V.

1. If we add another vector, ${\mathbf{x}}_{k+1}$, to the set, will we still have a spanning set? Explain.

2. If we delete one of the vectors, say, ${\mathbf{x}}_k$, from the set, will we still have a spanning set? Explain.

In ${ℝ}^{2\times 2}$, let

Show that $E_{11},E_{12},E_{21},E_{22}$ span ${ℝ}^{2\times 2}$.

Which of the sets that follow are spanning sets for $P_3$? Justify your answers.

1. $\{1,x^2,x^2-2\}$

2. $\{2,x^2,x,2x+3\}$

3. $\{x+2,x+1,x^2-1\}$

4. $\{x+2,x^2-1\}$

Let S be the vector space of infinite sequences defined in Exercise 15 of Section 3.1. Let $S_0$ be the set of $\{a_2\}$ with the property that $a_n\to 0$ as $n\to \infty$. Show that $S_0$ is a subspace of S.

Prove that if S is a subspace of ${ℝ}^1$, then either $S=\{\mathbf{0}\}$ or ${S=\mathbb{R}}^1$.

Let A be an $n\times n$ matrix. Prove that the following statements are equivalent:

1. $N(A)=\{\mathbf{0}\}$.

2. A is nonsingular.

3. For each $\mathbf{b}\in {\mathbb{R}}^n$, the system $A\mathbf{x}=\mathbf{b}$ has a unique solution.

Let U and V be subspaces of a vector space W. Prove that their intersection $U\cap V$ is also a subspace of W.

Let S be the subspace of ${ℝ}^2$ spanned by ${\mathbf{e}}_1$ and let T be the subspace of ${ℝ}^2$ spanned by ${\mathbf{e}}_2$. Is $S\cup T$ a subspace of ${ℝ}^2$? Explain.

Let U and V be subspaces of a vector space W. Define

Show that $U+V$ is a subspace of W.

Let S, T, and U be subspaces of a vector space V. We can form new subspaces using the operations of ∩ and + defined in Exercises 24 and 26. When we do arithmetic with numbers, we know that the operation of multiplication distributes over the operation of addition in the sense that

It is natural to ask whether similar distributive laws hold for the two operations with subspaces.

1. Does the intersection operation for subspaces distribute over the addition operation? That is, does

   $$S\cap (T+U)=(S\cap T)+(S\cap U)?$$

2. Does the addition operation for subspaces distribute over the intersection operation? That is, does

   $$S+(T\cap U)=(S+T)\cap (S+U)?$$