the second midterm

Location

In class, B102 during regular lecture time Thursday October 29

Topics

Vector space axioms

Examples of vector spaces: $\mathbb{F}^n$; $\mathcal{P}_n(\mathbb{F})$; $\mathcal{F}(S, \mathbb{F})$

Linear subspaces (definition&theorems: the intersection of subspaces is a subspace; a subspace is again a vector space)

Span of a set of vectors

Linear independence

Bases and Dimension

Linear Transformations

Format

The exam will have the following three kind of problems:

• Definitions&theorems. Be able to state them precisely.
• Short answer True/False questions
• A proof in which you provide more detail.
• A computational problem, using the concepts covered above.

Definitions and Theorems to know

You should be able to correctly state any of the following definitions, if asked, and any of the theorems if you use them.

Definition of a linear subspace: A subset $W\subset V$ is a linear subspace if - $W$ is not empty - $W$ is closed under vector addition - $W$ is closed under scalar multiplication

Theorem: If $W\subset V$ is a linear subspace then $W$ is a vector space.

Definition. The span of a set of vectors $S\subset V$ is the set of all linear combinations of vectors in $S$, i.e.

$\mathrm{Span}(S)=\bigl\{c_1u_1+\cdots+c_mu_m \mid m\in\N, c_1, \dots, c_m\in\mathbb{F}, u_1, \dots, u_m\in S\bigr\}$

Theorem. The span of a subset $S\subset V$ is a linear subspace of $V$.

Definition. The vectors $u_1, \dots, u_n\in V$ are linearly independent if for any $c_1, \dots, c_n\in \mathbb F$ with $c_1u_1+\cdots+c_nu_n =0$ one has $c_1=\dots=c_n=0$.

Theorem. If $u_1, \dots, u_n\in V$ are linearly independent and if one has

$a_1u_1+\cdots+a_nu_n = b_1u_1+\cdots+b_nu_n$

for certain $a_1, \dots, a_n, b_1, \dots, b_n\in \mathbb{F}$, then $a_1=b_1$, … , $a_n=b_n$.

Theorem. If $v\in\mathrm{span}(\{u_1, \dots, u_n\})$ then $\{v, u_1, u_2, \dots, u_n\}$ is linearly dependent.

Row reduction. Not a definition or theorem, but: you should know how to solve a system of $n$ linear equations with $m$ unknowns, where $n, m\leq 4$, using the method of row reduction, as done in lecture, or otherwise. In our course so far this comes up in the questions “does $x$ belong to the span of $\{u, v, w\}$?” or “Are the vectors $u_1, u_2, u_3$ linearly independent?”

Definition of basis. A set of vectors $\{u_1, \dots, u_n\}\subset V$ is a basis for $V$ if

• $\{u_1, \dots, u_n\}$ is linearly independent, and
• $\{u_1, \dots, u_n\}$ spans $V$.

Extension Theorem for Independent Sets. If $\{u_1, \dots, u_n\}\subset V$ is linearly independent, and $v\in V$ is not one the vectors $u_1, \dots, u_n$, then $v\in\mathrm{span}(u_1, \dots, u_n)$ if and only if $\{u_1, \dots, u_n, v\}$ is dependent.

Dimension Theorem. If $\{v_1, \dots, v_m\}\subset \mathrm{span}( u_1, \dots, u_n)$ and if $\{v_1, \dots, v_m\}$ is linearly independent then $m\leq n$.

Corollaries to the Dimension Theorem.

1. If $\{u_1, \dots, u_n\}$ and $\{v_1, \dots, v_m\}$ both are bases of a vector space $V$, then $m=n$.

2. If $L\subset V$ is a linear subspace and $V$ is finite dimensional then $\dim L\leq \dim V$. If $\dim L=\dim V$ then $L=V$.

3. If $L\subset V$ is a linear subspace and $V$ is finite dimensional, and if $\{v_1, \dots, v_k\}\subset L$ is a basis for $L$, then there exist vectors $v_{k+1}, \dots, v_n\in V$ such that $\{v_1, \dots, v_k, v_{k+1}, \dots, v_n\}$ is a basis for $V$.

Definitions.

• If a vector space $V$ has a basis $\{u_1, \dots, u_n\}$ with $n$ elements, then $n$ is the dimension of $V$.
• If a vector space $V$ has a basis with finitely many vectors then $V$ is called finite dimensional.

Definition. A map $T:V\to W$ from one vector space $V$ to another $W$ is called linear if for all $x, y\in V$ and all $a, b\in\mathbb{F}$ one has $T(ax+by) = aT(x)+bT(y)$.

Definition. If $T:V\to W$ is a linear map, then

• the Null space of $T$ is $N(T) = \{x\in V \mid T(x)=0\}$
• the Range of $T$ is $R(T) = \{T(x) \mid x\in V\} = \{y\in W \mid \exists x\in V: y=Tx\}$.

Theorem. The Null space a linear transformation $T:V\to W$ is linear subspace of $V$; the range of $T$ is a linear subspace of $W$.

Definition. The rank of $T$ is the dimension of the range of $T$.

Injectivity Theorem. A linear map $T:V\to W$ is injective if and only if $N(T)=\{0\}$.

Rank+Nullity Theorem. If $T:V\to W$ is linear, and if $V$ is finite dimensional, then

$\dim N(T) + \dim R(T) = \dim V .$

Bijectivity Theorem. If $V$ and $W$ are finite dimensional vector spaces with the same dimension, and if $T:V\to W$ is a linear transformation then the following are equivalent:

1. $T$ is injective (one-to-one)
2. $N(T)=\{0\}$
3. $\mathrm{rank}\,T = \dim V$
4. $T$ is surjective (onto)

Facts you can use

Unless you are asked to prove the theorem, you can use any of the theorems listed above without proving them. But do say “by the theorem that says…”

You can use that the sets $\mathbb{F}^n$, $\mathcal{P}_n(\mathbb{F})$, and $\mathcal{F}(S,\mathbb{F})$ with vector addition and scalar multiplication as defined in class are vector spaces. If you feel that justification is needed say “as is shown in the textbook…”