the first midterm

Location

In class, B102 during regular lecture time Tuesday October 6

Topics

Vector space axioms
Examples of vector spaces
- $\F^n$
- $\cP_n(\F)$ the space of polynomials of degree at most $n$ with coefficients in $\F$.
- $\cF(S, \F)$: the space of all functions $f:S\to\F$
Linear subspaces
- theorem: the intersection of subspaces is a subspace
- theorem: a subspace is again a vector space
Span of a set of vectors
- does $x\in\mathrm{Span}\{v_1,\dots,v_n\}$ ?
Linear independence
- Definition.
- Are $\{v_1, \dots, v_n\}$ linearly independent? In concrete examples in $\F^n$, or in $\cP_n(\F)$

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

Vector space axioms

Don’t remember them by number but by name. Here they are:

Vector addition is commutative and associative: for all $x,y, z\in V$ one has
- $x+y=y+x$
- $x+(y+z)=(x+y)+z$.
Existence of zero vector and additive inverse:
- there is a vector $0_V\in V$ such that for all $x\in V$ one has $x+0_V=x$.
- for every $x\in V$ there is a $y\in V$ such that $x+y=0_V$.
Multiplication with $1$: for all $x\in V$ one has $1_\F x=x$.
Associative and Distributive laws for scalar multiplication: For all $a,b\in\F$ and $x, y\in V$ one has
- $(ab)x = a(bx)$
- $a(x+y) = ax+ay$
- $(a+b)x = ax + bx$

Consequences of the axioms

Assume that $V$ is a vector space over a number field $\F$. You may assume that $\F=\R$, $\F=\C$, or $\F=\Q$. You should be able to prove the following directly from the axioms:

Prove there is only one zero vector
Prove that for each $x\in V$ there is only one additive inverse. This means that if $x+y=0$ and $x+z=0$ then $y=z$.
For every $x\in V$ one has $0_\F x=0_V $.
For every $x\in V$ an additive inverse is given by $(-1)x$.
For every $a\in\F$ one has $a 0_V = 0_V$.

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: If $u_1, \dots, u_n\in V$ and $c_1, \dots, c_n\in\F$ are given vectors and numbers, then the vector $x=c_1u_1+\cdots+c_nu_n$ is a linear combination of $u_1, \dots, u_n$.
Definition. The span of a set of vectors $S\subset V$ is the set of all linear combinations of vectors in $S$, i.e. \[ \mathrm{Span}(S)=\bigl\{c_1u_1+\cdots+c_mu_m \mid m\in\N, c_1, \dots, c_m\in\F, u_1, \dots, u_m\in S\bigr\} \]
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 $c_1, \dots, c_n\in F$ 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 $a_1, \dots, a_n, b_1, \dots, b_n\in \F$, 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?”

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 $\F^n$, $\cP_n(\F)$, and $\cF(S, \F)$ 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…”