MATH 3210 Exam 1 In-Class Review

notevector-spaces

Question 1

A vector space is a set $V$ with addition and scalar multiplication such that,

• Addition is commutative.
• There exists an additive identity ($0$).
• There exists a unique additive identity, $-v$, for all $v\in V$.
• Both addition and scalar multiplication are associative.
• Distribution laws are respected.

Question 2

Show $\mathcal{P}(\mathbb{R})$ satisfies the vector space axioms above.

Question 3

Show ${\mathcal{P}}_m(\mathbb{R})$ is a subspace, $0\in {\mathcal{P}}_m(\mathbb{R})$, and it’s closed under addition and scalar multiplication.

Question 4

We have that $V_1+V_2=\left\{v_1+v_2:v_1\in V_1,v_2\in V_2\right\}$. And we define $V_1\oplus V_2$ as $V_1+V_2$ if there is a unique way to write any element of $V_1+V_2$ as $v_1+v_2$.

Question 5

Does $V_1\oplus U=V_2\oplus U⟹V_1=V_2$? No?

Question 6

Question 7

Show $\mathrm{span}(v_1,\dots ,v_n)$ satisfies the necessary subspace conditions.

Question 8

Let $u_1,\dots ,u_m$ be a list. If the list is linearly dependent, then there exists some $k$, $1\le k\le m$ where $u_k\in \mathrm{span}(u_1,\dots ,u_{k-1})$ and $\mathrm{span}(u_1,\dots ,u_m)=\mathrm{span}(u_1,\dots ,u_{k-1},u_{k+1},\dots u_m)$.

Question 9

Find which of $(1,2,3),(7,1,4),(0,1,1),(7,6,9)$ can be written as a sum of the other vectors.

$(1,2,3)$ linearly independent $(1,2,3),(7,1,4)$ linearly independent $(1,2,3),(7,1,4),(0,1,1)$ seem to be linearly independent $(1,2,3),(7,1,4),(0,1,1),(7,6,9)$ can’t be linearly independent since a space of dimension 3 can have a list of at most 3 linearly independent vectors.

Note: if you are unsure, you can put the vectors into a matrix and do gaussian elimination to see if we can get to reduced row echelon form.

Question 10

Extend $(1,1,0),(0,1,1)$ to a basis of ${\mathbb{R}}^3$.

Add to the list a set of vectors that themselves form a basis of ${\mathbb{R}}^3$, so the list spans ${\mathbb{R}}^3$: $(1,1,0),(0,1,1),(1,0,0),(0,1,0),(0,0,1)$.

Now use the linear dependence lemma to remove vectors that are in the span of the rest of the vectors, which leaves us with $(1,1,0),(0,1,1),(1,0,0)$ in this case.

Question 11

As we’ve shown, since ${\mathbb{F}}^3$ can be spanned by a linearly independent set of three vectors, the dimension of ${\mathbb{F}}^3$ is three.

Question 12

A vector space $T:V\to W$ is a morphism from a vector space $V$ to a vector space $W$ that respects sums and products in $V$ in $W$ (additivity and homogeneity).

Question 13

Only $b=0$ works in this case because any non-zero value of $b$ causes the additivity condition to not be satisfied for $T$.

For $T(a)=2a+b$, to be a linear map, we need that $T(a_1+a_2)=T(a_1)+T(a_2)$ or equivalently $2(a_1+a_2)+b=2a_1+b+2a_2+b=2(a_1+a_2)+2b$, which is only true if $b=0$.

Question 14

The nullspace for $T:V\to W$ is the subset $N\subseteq T$ such that for $n\in N$, $T(n)=0_W$.

The range for $T:V\to W$ is the subset $R\subseteq W$ such that for $r\in R$, there exists $v\in V$ such that $T(v)=r$.

Question 15

1. Multiplication by a constant $c\in \mathbb{R}$.
2. The differentiation map.
3. The identity map.