Linear Algebra

A Vector Space \(V\) is defined over a field \(F\) is a non-empty set that has two operations: 1. Vector Addition: \(+:V \times V \to V\) 2. Scalar Multiplication: \(\cdot:F \times V \to V\)

Satisfying the following properties for all \(u,v,w \in V\) and \(\alpha,\beta \in F\):

1. Closure: \(u+v \in V\) and \(\alpha \cdot v \in V\)
2. Commutativity (addition): \(u+v = v+u\)
3. Associativity: \(u+(v+w) = (u+v)+w\)
4. Zero Vector: \(\exists 0 \in V\) such that \(0+v=v\)
5. Additive Inverse: \(\exists -v \in V\) such that \(v+(-v)=0\)
6. Commutativity (multiplication): \(\alpha\cdot(\beta\cdot v)= (\alpha\times\beta)\cdot v\)
7. Field Identity preserves vector: \(1\cdot v = v\)
8. Distributivity: \(\alpha \cdot (u+v) = \alpha\cdot u + \alpha \cdot v\)

Vector Subspaces

A Subspace is a subset of a vector space that is itself a vector space. Properties 2, 3,

A set of nonzero vectors \(v_1, \ldots, v_n\) in a vector space \(V\) over field \(F\) is linearly dependent if there exists some set of field elements \(\alpha_1, \ldots, \alpha_n\) not all zero such that: \[ 0 = \sum_{i=1}^n \alpha_i \cdot v_i \] If no such set of \(\alpha_i\) exists then the set of vectors is linearly independent.

Let \(u,v,w \in V\), a vector space over field \(F\). A function, \(<*,*>:V \times V \to F\) is called an inner product if:

1. \(<u+v,w> \;=\; <u,w> + <v,w>\)
2. \(<\alpha \cdot u,w> \;=\; \alpha <u,w>\)
3. \(<u,v> \;=\; \overline{<v,u>}\), (where \(\overline{\alpha}\) is the complex conjucate of \(\alpha\))
4. \(<u,u> \;\geq\; 0\) with equality iff \(u=0\)

Orthogonality

Two nonzero vectors \(u\) and \(v\) are orthogonal if \(<u,v\)=0$

Consequences

• Orthogonal vectors are also linearly independent

A function \(||\;|| : V \to \mathbb{R}^+\) satisfying:

1. \(||\alpha \cdot v|| = |\alpha| \cdot ||v||\)
2. \(||v|| \geq 0\) with equality iff \(v=0\)
3. \(||u+v|| \leq ||u||+||v||\)

\[A^HA = \mathbb(1) = AA^H\]

Right Inverses

Left Inverse

Inverse

Pseudo Inverse