Linear algebra starts not with matrices but with axioms: a field of scalars and a vector space over it. Everything later — bases, dimension, eigenvalues — is a consequence of these rules, which is why the first proofs in the course are about the axioms themselves.
Defining a vector space abstractly — by axioms, not as arrows or columns — means one theorem covers polynomials, functions, and data vectors at once. Abstraction buys reach.
Vector spaces over $\mathbb{R}$ and $\mathbb{C}$ are the language of machine learning (data as vectors), quantum mechanics, and signal processing; finite fields power error-correcting codes and cryptography.
Grassmann and Peano formalized vector spaces in the late 1800s; the abstract axiomatic view took hold in the 20th century and is now the default language of pure and applied mathematics alike.
A field $F$ is a set with two operations $+$ and $\cdot$ such that both are associative and commutative, distributivity holds, there are identities $0\ne1$, every element has an additive inverse, and every nonzero element has a multiplicative inverse. Examples: $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$, and the integers mod $p$, written $\mathbb{Z}_p$, for prime $p$.
A vector space over a field $F$ is a set $V$ with vector addition and scalar multiplication satisfying, for all $u,v,w\in V$ and $a,b\in F$: (VS1) $u+v=v+u$; (VS2) $(u+v)+w=u+(v+w)$; (VS3) a zero vector with $v+0=v$; (VS4) additive inverses; (VS5) $1v=v$; (VS6) $(ab)v=a(bv)$; (VS7) $a(u+v)=au+av$; (VS8) $(a+b)v=av+bv$.
$F^n$ (coordinate tuples), the $m\times n$ matrices $M_{m\times n}(F)$, the polynomials $P(F)$, and the functions from any set $S$ into $F$ — with pointwise operations — are all vector spaces over $F$. So is $\mathbb{C}$ as a vector space over $\mathbb{R}$: the field of scalars is part of the data.
If $u+w=v+w$ in a vector space, then $u=v$.
For every $v\in V$ and $a\in F$: $0v=0$, $a\,0=0$, $(-1)v=-v$, and $av=0$ implies $a=0$ or $v=0$.
• Closure under both operations must be checked first — it is not an
axiom you get for free.
• The zero vector must actually lie in the set; a set missing $0$ is
disqualified immediately.
• Scalars come from the field: over $\mathbb{Z}_p$ arithmetic wraps around, and
facts like "$2\ne0$" can fail (in $\mathbb{Z}_2$, $2=0$).
• $0v=0$ and $(-1)v=-v$ are theorems, not axioms — a proof may not assume
them before proving them.
Original problems written for this guide in the style of Friedberg, Insel & Spence exercises. Write a full solution before opening the answer.
Is $\{(x,y):x+y=0\}\subseteq\mathbb{R}^2$, with the usual operations, a vector space?
Is $\{(x,y):x+y=1\}$ a vector space?
Compute $3\cdot4$ and $3+4$ in $\mathbb{Z}_5$.
Prove from the axioms that $0v=0$ for every vector $v$.
Prove $(-1)v=-v$, i.e. that $(-1)v$ is the additive inverse of $v$.
Find $4^{-1}$ in $\mathbb{Z}_7$.
Show that $\mathbb{Z}_6$ is not a field.
Explain why $\mathbb{R}$ is a vector space over $\mathbb{Q}$, and why $\mathbb{Q}$ is not a vector space over $\mathbb{R}$ with the usual operations.
Course coverage. This unit is part of the complete MAT240H1 topic map. It should cover the following items before you move on:
Use the current UofT MAT240H1 calendar topics as the coverage spine, and pair lecture notes with a Friedberg-Insel-Spence-style linear algebra text for extra reading. The exact assigned textbook should still be checked against the instructor syllabus.
The algebra spine is: choose the field, define addition and scalar multiplication, verify closure and axioms, then reason about vectors abstractly.
Use these after the main practice set. They are designed to force definition-level reasoning, theorem selection, and mixed application rather than routine calculation only.
Why is $\mathbb{Z}_p$ a field when $p$ is prime?
Show the set of polynomials of degree at most $3$ over $F$ is a vector space.
Why are polynomials of exactly degree $3$ not a vector space?
Compare $\mathbb{C}$ as a vector space over $\mathbb{C}$ and over $\mathbb{R}$.