9 Linear Algebra
9.1 Elementary Example
9.1.1 Simple
Linear algebra studies vectors and linear maps. Here three vectors in a plane.
\[ V = \{ e_{1},\ e_{2},\ v \} \]
\[ e_{1} = (1,0),\quad e_{2} = (0,1),\quad v = (2,3) \]
where
- \(V\) is a set of sample vectors.
- \(e_{1}, e_{2}\) form the standard basis of \(\mathbb{R}^{2}\).
9.1.2 General
Matrices represent linear maps. A \(3 \times 3\) matrix acts on \(\mathbb{R}^{3}\).
\[ A = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix} \]
\[ A(x_{1},x_{2},x_{3}) = (x_{1},\ 2 x_{2},\ 3 x_{3}) \]
where
- \(A\) is a linear transformation written as a matrix.
- Adjoint
- Basis
- Characteristic Polynomial
- Complex Conjugate
- Conjugate Symmetry
- Conjugate Transpose
- Determinant
- Diag
- Dual Space
- Eigendecomposition
- Eigenvalue
- Eigenvector
- Hermitian
- Hermitian Conjugate
- Homogeneity
- Homogeneous
- Inner Product
- Inner Product Space
- Kronecker Delta
- Linear Function
- Linear Map
- Linear Transformation
- Operator
- Orthonormal
- Orthonormal Set of Functions
- Projection Map
- Quadratic
- Standard Basis