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Vector Spaces
Suppose we have a set $V$ 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. $\begin{bmatrix}
3 \\ 2 \end{bmatrix}$, $\begin{bmatrix} 0 \\ 0 \end{bmatrix}$, $\begin{bmatrix} \pi \\ e \end{bmatrix}$, ...
- there's a picture that goes with $\mathbb R^2$
-
- so, we can picture every vector in the space
- (same for $\mathbb R^3$)
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.
- Inner Product: e.g. $\langle A, B \rangle = \sum_{ij} a_{ij} b_{ij}$
- norm: e.g. Frobenius Norm: $\| A \|_F = \langle A, A \rangle$
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
- Linear Algebra MIT 18.06 (OCW)
- Курош А.Г. Курс Высшей Алгебры