ML Wiki

Vector Spaces

Suppose we have a set of vectors $V$ and elements $\mathbf v_1, ..., \mathbf v_i ... \in V$

• we define addition on $V$ where we map any pair $\mathbf v_i, \mathbf v_j \in V$ to a value $\mathbf v_i + \mathbf v_j$
• and we define the operation scalar multiplication where for any scalar number $c$ and a vector $\mathbf v \in V$ we have a value $c \cdot \mathbf v$

So, what can we do with elements in a vector space?

• multiply them by a scalar
• it means we should be able to take linear combinations of elements in the space

Axioms

The elements of $V$ are vectors and $V$ is a space if the axioms hold

• commutativity: $\mathbf v_i + \mathbf v_j = \mathbf v_j + \mathbf v_i$
• associativity: $(\mathbf v_i + \mathbf v_j) + \mathbf v_k = \mathbf v_j + (\mathbf v_i + \mathbf v_k)$
• there exists an element $\mathbf 0 \in V$ s.t. $\mathbf 0 + \mathbf v = \mathbf v$
• for any element $\mathbf v$ there exists the opposite $-\mathbf v$ s.t. $\mathbf v + (-\mathbf v) = \mathbf 0$
• therefore can define difference as $\mathbf v_1 - \mathbf v_2 = \mathbf v_1 + (-\mathbf v_2)$

multiplication on scalars ($c$'s are scalars):

• $c\, (\mathbf v_1 + \mathbf v_2) = c\, \mathbf v_1 + c\, \mathbf v_2$
• $(c_1 + c_2)\, \mathbf v = c_1 \mathbf v + c_2 \mathbf v$
• $(c_1 \cdot c_2) \cdot \mathbf v = c_1 \cdot (c_2 \cdot \mathbf v)$
• $1 \cdot \mathbf v = \mathbf v$

Implications:

• $c \cdot \mathbf 0 = \mathbf 0$
• $0 \cdot \mathbf v = \mathbf 0$
• if $c \cdot \mathbf v = \mathbf 0$ then either $c = 0$ or $\mathbf v = \mathbf 0$
• $c \cdot (- \mathbf v) = - c \cdot \mathbf v$
• $(- c) \cdot \mathbf v = - c \cdot \mathbf v$
• $c\, (\mathbf v_1 - \mathbf v_2) = c\, \mathbf v_1 - c\, \mathbf v_2$
• $(c_1 - c_2)\, \mathbf v = c_1 \mathbf v - c_2 \mathbf v$

Example: Coordinate Spaces

• $\mathbb R^2$ - real numbers ("$x/y$ plane")
• e.g. $\begin{bmatrix} 3 \\ 2 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 0 \end{bmatrix}$, $\begin{bmatrix} \pi \\ e \end{bmatrix}$, ...
• there's a picture that goes with $\mathbb R^2$
• so, we can picture every vector in the space
• (same for $\mathbb R^3$)

Linear Span

A linear span (or just span) of a set of vectors $V = \{ \mathbf v_1, ..., \mathbf v_n \}$

• is a set of all linear combinations of these vectors:
• $\text{span}(V) = \{ \sum \beta_j \mathbf v_i \ \forall \beta_j \in \mathbb R \}$
• Linear span of $V$ is a Vector Space

Unique representation

• if vectors of $V$ are linearly independent and $\mathbf b \in V$
• then $\mathbf b$ is a unique linear combinations of vectors from $V$
• i.e. $\mathbf b = \sum \beta_j \mathbf v_i$ and all $\beta_j$ are unique

Basis

Maximal Independent Subset

• if $V^*$ is maximal independent subset of $V$ (all vectors in $V$ are linearly independent and $V^*$ is not contained in any other subset of linearly independent vectors)
• then $\text{span}(V) = \text{span}(V^*)$
• and $V^*$ is the basis for $\text{span}(V)$

Vector Subspaces

A subspace of a vector space should form a space on it's own.

Any line through the origin:

• is it a vector space?
• yes. We can take any scalar, and the result will still be on the line
• if the line is not through the origin, then multiplying by 0 will bring us out of the space - so the origin must be included

For a Matrix there are Four Fundamental Subspaces:

Vector Spaces

Matrix Vector Spaces

A matrix space is also a vector space, where elements are matrices of the same dimensionality: we can multiply matrices by a scalar and can add two matrices of the same dimension.

Function Spaces

In a function space, the "vectors" are functions:

• we can define an Inner Product as $\langle f, g \rangle = \int\limits_{-\infty}^{\infty} f(x) \, g(x) \, dx$ with Integral instead of sum
• and we define orthogonality as $\langle f, g \rangle = 0$