In Exercise 1 of Section 3.3, indicate whether the given vectors form a basis for ${ℝ}^2$.

In Exercise 2 of Section 3.3, indicate whether the given vectors form a basis for ${ℝ}^3$.

Consider the vectors

1. Show that ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$ form a basis for ${ℝ}^2$.

2. Why must ${\mathbf{x}}_1$, ${\mathbf{x}}_2$, ${\mathbf{x}}_3$ be linearly dependent?

3. What is the dimension of $\text{Span}({\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3)$?

Given the vectors

what is the dimension of $\text{Span}({\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3)$?

Let

1. Show that ${\mathbf{x}}_1$, ${\mathbf{x}}_2$, and ${\mathbf{x}}_3$ are linearly dependent.

2. Show that ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$ are linearly independent.

3. What is the dimension of $\text{Span}({\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3)$?

4. Give a geometric description of $\text{Span}({\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3)$.

In Exercise 2 of Section 3.2, some of the sets formed subspaces of ${ℝ}^3$. In each of these cases, find a basis for the subspace and determine its dimension.

Find a basis for the subspace S of ${ℝ}^4$ consisting of all vectors of the form ${(a+b,a-b+2c,b,c)}^T$, where a, b, and c are all real numbers. What is the dimension of S?

Given ${\mathbf{x}}_1=(1,1,1{)}^T$ and ${\mathbf{x}}_2=(3,-1,4{)}^T$:

1. Do ${\mathbf{x}}_1$ and ${\mathbf{x}}_2$ span ${ℝ}^3$? Explain.

2. Let ${\mathbf{x}}_3$ be a third vector in ${ℝ}^3$ and set $X=\left({\mathbf{x}}_1{\mathbf{x}}_2{\mathbf{x}}_3\right)$. What condition(s) would X have to satisfy in order for ${\mathbf{x}}_1$, ${\mathbf{x}}_2$, and ${\mathbf{x}}_3$ to form a basis for ${ℝ}^3$?

3. Find a third vector ${\mathbf{x}}_3$ that will extend the set {${\mathbf{x}}_1$, ${\mathbf{x}}_2$} to a basis for ${ℝ}^3$.

Let ${\mathbf{a}}_1$ and ${\mathbf{a}}_2$ be linearly independent vectors in ${ℝ}^3$, and let x beavector in ${ℝ}^2$.

1. Describe geometrically $\text{Span}({\mathbf{a}}_1,{\mathbf{a}}_2)$.

2. If $A=({\mathbf{a}}_1,{\mathbf{a}}_2)$ and $\mathbf{b}=A\mathbf{x}$, then what is the dimension of $\text{Span}({\mathbf{a}}_1,{\mathbf{a}}_2,\mathbf{b})$? Explain.

The vectors

span ${ℝ}^3$. Pare down the set $\{{\mathbf{x}}_1,{\mathbf{x}}_2,{\mathbf{x}}_3,{\mathbf{x}}_4,{\mathbf{x}}_5\}$ to form a basis for ${ℝ}^3$.

Let S be the subspace of $P_3$ consisting of all polynomials of the form ${\mathit{\text{ax}}}^2+\mathit{\text{bx}}+2a+3b$. Find a basis for S.

In Exercise 3 of Section 3.2, some of the sets formed subspaces of ${ℝ}^{2\times 2}$. In each of these cases, find a basis for the subspace and determine its dimension.

In $C[-\pi ,\pi ]$, find the dimension of the subspace spanned by $1,\text{cos }2x,{\text{cos}}^{2}x$.

In each of the following, find the dimension of the subspace of $P_3$ spanned by the given vectors:

1. $x,x-1,x^2+1$

2. $x,x-1,x^2+1,x^2-1$

3. $x^2,x^2-x-1,x+1$

4. $2x,x-2$

Let S be the subspace of $P_3$ consisting of all polynomials $p\left(x\right)$ such that $p\left(0\right)=0$, and let T be the subspace of all polynomials $q\left(x\right)$ such that $q\left(1\right)=0$. Find bases for

1. S

2. T

3. $S\cap T$

In ${ℝ}^4$, let U be the subspace of all vectors of the form ${(u_1,u_2,0,0)}^T$, and let V be the subspace of all vectors of the form ${(0,v_2,v_3,0)}^T$. What are the dimensions of $U,V\mathrm{.}U\cap V,U+V$? Find a basis for each of these four subspaces. (See Exercises 24 and 26 of Section 3.2.)

Is it possible to find a pair of two-dimensional subspaces U and V of ${ℝ}^3$ whose intersection is {0}? Prove your answer. Give a geometrical interpretation of your conclusion. [Hint: Let $\{{\mathbf{u}}_1,{\mathbf{u}}_2\}$ and $\{{\mathbf{v}}_1,{\mathbf{v}}_2\}$ be bases for U and V, respectively. Show that ${\mathbf{u}}_1,{\mathbf{u}}_2,{\mathbf{v}}_1,{\mathbf{v}}_2$ are linearly dependent.]

Show that if U and V are subspaces of ${ℝ}^n$ and $U\cap V=\left\{\mathbf{0}\right\}$, then