Abstract Linear Algebra
#print vector_space
vector_space (R: Type u) (M: Type v) [field R] [add_comm_group M] := semimodule R Mvector space is a
- semi-module over a
- field R
- additive communitative group M ```plantuml
```
0.1 Direct Sum
Find two unrelated spaces in your universe and slam them together
Given some vector space \(S\), we choose some subspaces \(\{X,Y,Z...\}\)
- None of these subspaces share any element but the null vector
- there is no combination of \(\{Y,Z...\}\) that can form an element of \(X\).
take an element from each subspace in \(\{X,Y,Z...\}\) then add them.
\[\oplus \{X,Y,Z...\} = \{x + y + ... | x \in X, y \in Y ...\}\]
0.2 Direct Sum of matrices
Given that \(A\) and \(B\) are matrices themselves.
Note that \(0\) stands for zero matrices of varying sizes.
Block matrix notation of direct sum:
\[A \oplus B = \begin{bmatrix} A & 0 \\ 0 & B \end{bmatrix}\]
\[B \oplus A = \begin{bmatrix} B & 0 \\ 0 & A \end{bmatrix}\]
\[ A \oplus B \cong B \oplus A \]
- Direct sum is commutative up to isomorphism
0.2.1 Example
\(\{(x,0) | x\in \mathbb{R}\} \oplus \{(0,y) | x\in \mathbb{R}\}=\mathbb{R}^2\) or written as
\((\mathbb{R},0) \oplus (0,\mathbb{R})\)
but some might even do notational abuse and just write
\(\mathbb{R} \oplus \mathbb{R}\)
1 Linear Map
\[ T \in \mathcal{L}(V,W) \]
T :: V -> W
2 StoryTime
Vector space of cars Field of colors Add means “mix”
Red car mix Blue car can be exchanged for a purple Car and vice-versa.
Vector space of cars equivalent to saying a magic rainbow brush that can paint any car.
- Basis vectorspace: cars of fundamental colors.
- Zero-Vector: The no-car when painted with any color still results in the no-car.
- Inverse: Green car mix Red car makes the no-car. Green is known as anti-red.
Linear Map T :: Cars -> Trucks
We can exchange our painted cars for painted trucks.
T is linear map if
- we can trade a red car for a red truck, blue car blue truck, etc.
- we can trade “red car mix blue car” for a “purple truck”
Proof: Dim Range + Dim Null
- red car trades for purple truck
- yellow car trade for purple truck
- red car mix anti-yellow car trades for no-truck.
- Null space = red mix anti-yellow car
- Range = purple truck
Notice how the 2 cars collapse into the 1 purple truck in the range
is given back through the 1 “red mix anti-yellow car” in the null space
3 Spectral Theorem
- Hermitian operator
- For Real vector space its symmetric matrix
- For Complex vector space its symmetric complex conjugate matrix
- For Infinite dim vector space its Complex self adjoint
- Symmetric matrix means it looks the same when transposed
- Typically a random matrix has complex and real eigenvalues regardless of elements are only real
- Spectral Thm: A Hermitian(symmetric) matrix that can have complex or real elements will always have real eigenvalues
- Eigenvectors are orthogonal
4 2 by 2 matrix
4.1 Positive definite matrices
Given a symmetric 2 by 2 matrix, four ways to tell if positive definite:
- Eigenvalue test: \(\lambda_1 \gt 0, \lambda_2 \gt 0\)
- Determinants test: \(a \gt 0, ac - b^2 \gt 0\)
- Pivot test: \(a \gt 0, \frac{ac-b^2}{a} \gt 0\)
- \(x^TAx\) is positive definite except when \(x=0\) (the actual definition of positive definite)
Note that \(x^TAx\) results in an eqn like \(2x_1^2+12x_1x_2+18x_2^2\) and if this eqn is postive for all real \(x_1, x_2\) then the matrix \(A\) is postive definite.
