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?

  • add two elements
  • 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. [math]\begin{bmatrix} 3 \\ 2 \end{bmatrix}[/math], [math]\begin{bmatrix} 0 \\ 0 \end{bmatrix}[/math], [math]\begin{bmatrix} \pi \\ e \end{bmatrix}[/math], ...
  • there's a picture that goes with $\mathbb R^2$
  • 774a1e4efbfb4ee9996aa4a14d184659.png
  • 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:

  • 36680970ea4e49dd8690c9ae3b9f8e84.png
  • 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 $\langle f, g \rangle = 0$


Sources

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