Linear Algebra Primer

References

Definitions

Linear Combination

A Linear Combination of vectors is the vector of form

$$a_1v_1 + a_2v_2 + a_3v_3 + \dots + a_kv_k$$

Span

The set of all possible linear combinations of a set of vectors.

Linear Independence

A set of vectors is linearly independent if no vector in the list is in the span of the others.

For example, the set of vectors $(1, 0, 0), (0, 1, 0), (0, 0, 1)$ are linearly independent. Specifically, $(0, 0, 1)$ cannot be written as any linear combination of the other vectors. That is, given $a, b \in F$, $a(1, 0, 0) + b(0, 1, 0) = (a, b, 0)$.

Basis

A list of vectors $(v_1, v_2, \dots, v_n)$ is a basis for $V$ if its elements are linearly independent, and their span is (the whole vector space) $V$. The dimension of a vector space is the length of any basis.

Subspaces

Linear maps

A function $f : V \to W$ is a linear map if it preserves the operations of addition and scalar multiplication. That is, for all $v, w \in V, c \in F, f(v+w) = f(v) + f(w)$ and $f(cv) = cf(v)$.

Theorems

Every vector $v \in V$ is a linear combination of the basis vectors of $V$.

Proof:

Let $v$ be an arbitrary vector in $\mathbb{R}^3$ and $e_1 = (1, 0, 0), e_2 = (0, 1, 0), e_3 = (0, 0, 1)$ the basis vectors of $\mathbb{R}^3$. Then:

$$ (a_1,a_2,a_3) = (a_1, 0, 0) + (0, a_2, 0) + (0, 0, a_3) = a_1(1, 0, 0) + a_2(0, 1, 0) + a_3(0, 0, 1) = a_1e_1 + a_2e_2 + a_3e_3 $$

For all finite dimensional vector spaces all bases have the same length.

Proof:

Missing proof

Every linear map sends $0$ to $0$.

Proof:

Let $V, W$ be vector spaces and $f: V \to W$ a linear map. For all $v \in V$:

$$f(0 + v) = f(v) = f(0) + f(v) = 0 + f(v)$$

Every linear map sends additive inverses to additive inverses.

Proof:

Let $V, W$ be vector spaces and $f: V \to W$ a linear map. For all $v \in V$:

$$f(v + (-v)) = f(v) + f(-v) = f(0)$$