Exercises

Let $V$ be a vector space over a field $F$. A student is asked to prove that for any $\alpha \in F$, it is always true that $\alpha \overrightarrow{0}=\overrightarrow{0}$.

[Student box]

For any $\alpha \in F$, we know

Therefore, $\alpha \overrightarrow{0}=\overrightarrow{0}$, as required.

[End of Student box]

This proof only works when $V=F^n$. Can you find a proof that works for any vector space $V$? [Hint: Use techniques from the proof of Lemma 2.5(iii).]

Consider $(1,-1,1),(-3,-5,7),$ and $(3,1,-2)$ in ${ℝ}^3$.

1. i.

   Prove this sequence of three vectors is linearly dependent.

2. ii.

   Express each of the vectors as a linear combination of the other two.

Prove that .

Are these subspaces of ${ℝ}^3$? Justify your answer, and when it is a subspace, find a basis.

1. i.

   $W:=\{(x,y,z)\in {ℝ}^3|x+y+z=1\}$.

2. ii.

   $W:=\{(x,y,z)\in {ℝ}^3|x-y=z\}$.

3. iii.

   $W:=\{(x,y,z)\in {ℝ}^3|\mathrm{\hspace{0.33em}2}x+y-3z=0\}$.

4. iv.

   $W:=\{(x,y,z)\in {ℝ}^3|y^2=2x\}$.

5. v.

   $W:=\{(x,y,z)\in {ℝ}^3|x+2y=y-z=0\}$.

Consider the set of polynomials $W:=\{f\in {𝒫}_3(ℝ)|f(1)=0\}$.

1. i.

   Prove that $W$ is a subspace of ${𝒫}_3(ℝ)$.

2. ii.

   Choose three polynomials in $W$ of degrees 1, 2, and 3 respectively. Prove that your three polynomials form a basis of $W$.

Prove that, in any vector space $V$, if a sequence of vectors includes the zero vector, then the sequence is linearly dependent.

For the following subspaces, find a basis, and hence the dimension.

1. i.

   $W:=\mathrm{span}\{(1,0,-1,1),(0,1,-3,2),(-1,2,0,1),(0,4,0,-1)\}\subset {ℝ}^4$

2. ii.

   The symmetric matrices in ${\mathrm{M}}_3(ℝ)$.

3. iii.

   $W:={\mathrm{span}}_{ℝ}\{1+i,1-i,2+3i\}\subset ℂ$ as a vector space over $ℝ$.

4. iv.

   $W:=\mathrm{span}\{(1,2,3),(1,-1,0),(2,1,3)\}$ of ${ℝ}^3$.

5. v.

   $W:=\{(x,y,z,w)|x+y+z=y-w=0\}$ of ${ℝ}^4$.

Prove that if $ℬ=(\overrightarrow{x_1},\mathrm{\cdots },\overrightarrow{x_n})$ is a sequence of vectors such that some vector is repeated in the sequence, then $ℬ$ is linearly dependent.

Let $W\subset V$ be a subspace such that $dimW=dimV$. Prove $W=V$.

Let $V$ be an $n$-dimensional vector space over a field $F$. A student is asked to prove Theorem 2.38(ii), which says that if a set of $n$ vectors spans $V$, then they must form a basis. He is also asked to state every theorem that he uses. His proof goes as follows:

[Student box]

Assume $\overrightarrow{v_1},\mathrm{\cdots },\overrightarrow{v_n}$ spans $V$. By Theorem 2.26 we can choose a subset of these vectors of size $m$ which forms a basis of $V$. Since every basis of $V$ has dimension $n$ by Corollary 2.37, we must have $m=n$. In other words, the subset is the whole set, and so $\overrightarrow{v_1},\mathrm{\cdots },\overrightarrow{v_n}$ is a basis. QED.

[End of Student box]

What has the student done wrong, and how might he get full marks?

Determine whether or not the following sets $V$ are vector spaces over the given field $F$:

1. i.

   Let $V=\mathbb{Q}+\mathbb{Q}\sqrt{2}=\{a+b\sqrt{2}|a,b\in \mathbb{Q}\}$ and $F=\mathbb{Q}$, with the usual addition and scalar multiplication.

2. ii.

   Given a non-empty set $S$, let $V$ be the set of functions from $S$ to a field $F$. Addition and scalar multiplication are defined as in Example 2.1(vi).

3. iii.

   Let $V=\{x\in ℝ|x>0\}$, and $F=ℝ$. Define a new “addition” to be $x\oplus y:=xy$, and use the usual scalar multiplication in $ℝ$. We introduced a different symbol for the new addition, to avoid confusion with the “usual” addition in $ℝ$.

Consider the set of all polynomials over a field $F$:

This includes polynomials of arbitrarily large degree, unlike ${𝒫}_n(F)$, which only takes polynomials of degree less than or equal to $n$. You may assume $𝒫(F)$ forms a vector space over $F$ under the usual addition and scalar multiplication. It is not possible to find a basis for $𝒫(F)$ which consists of a finite number of vectors. Why not?

Notice that . Is it possible to write any matrix $A\in {\mathrm{M}}_2(ℝ)$ as a linear combination of a symmetric matrix and a skew-symmetric matrix? Justify your answer.

Is it possible to write any matrix $A\in {\mathrm{M}}_n({𝔽}_2)$ as a linear combination of a symmetric matrix and a skew-symmetric matrix? Justify your answer.

[Recall the field ${𝔽}_2$ is the two element field from Example 1.1(iii).]

Let $V$ be the subspace of functions $ℝ\to ℝ$ given by

1. i.

   The zero vector $\overrightarrow{0}\in V$ is a function $ℝ\to ℝ$. Draw, or describe, that function.

2. ii.

   Prove that $T=\{f\in V|f(0)=2\}$ is not a subspace of $V$.

3. iii.

   Prove that $S=\{f\in V|f(x)=a(e^x-e^{-x}),a\in ℝ\}$ is a subspace of $V$.

This example is common in analysis modules, such as MATH317. Let $l^{\mathrm{\infty }}([0,1])$ (pronounced “Little ell infinity”) be the set of all bounded real functions whose domain is the interval $[0,1]$, and whose codomain is $ℝ$. In other words, the set of all $f$ for which there exists an $M>0$ such that , for every $0\le x\le 1$. It is true, and you may assume that $l^{\mathrm{\infty }}([0,1])$ is a vector space over $ℝ$, where addition and scalar multiplication are defined as in Example 2.1(vi). Which of the following subsets are subspaces?

1. i.

   The set of continuous functions from $[0,1]$ to $ℝ$.

2. ii.

   $\{f\in l^{\mathrm{\infty }}([0,1])|f(1)=1\}$.

3. iii.

   $\{f\in l^{\mathrm{\infty }}([0,1])|f(x)=f(0)\text{ for all }x\in [0,1]\}$.

Consider the set of formal power series over a field $F$:

So, $F[[x]]$ includes all of the polynomials over $F$ (compare with Exercise 2.65), but also much more. For example, the formal power series $1+x+x^2+x^3+\mathrm{\cdots }$ is not a polynomial, because it has infinitely many non-zero coefficients; but it is in $F[[x]]$. Also, the Taylor series of $\sin (x)$ is in $\mathbb{Q}[[x]]$, but it’s not a polynomial.

Is $ℝ[[x]]$ a vector space over $ℝ$? Justify your answer.