Linear Algebra 15 | Linear Algebra Cheatsheet
Linear Algebra 15 | Linear Algebra Cheatsheet
- Vector
- Dimension
- Linear Combination
- Dot Product
- Outer Product
- Norm
- Scaling
- Rotation
- Projection
- Normalization (Unity)
- Perpendicular
- Schwarz inequality
- Triangle inequality
2. Matrix
- Matrix times vector
- Associative law
- No commutative law
- Pivots
- Identity Matrix
- Elimination Matrix
- Inverse Elimination Matrix
- Exchange Matrix
- Augment Matrix
- Transpose Matrix
- Inverse Matrix
- Transpose Rule
- Inverse Rule
- Invertible Conditions
(1) Pivots: A is invertible if and only if it has n pivots.
(2) Linear Independence: A is invertible if and only if the column vectors of A are all linear independent.
(3) Determinate: A is invertible if and only if det(A) ≠ 0.
(4) Eigenvalues: A is invertible if and only if all the eigenvalues of A ≠ 0.
(5) Definite: If A is a semi-definite symmetric matrix, then A is invertible.
(6) Nullspace: A is invertible if and only if N(A) = ∅.
(7) Diagonally Dominant: If A is a diagonally dominant matrix, then A is invertible.
(8) etc
- LU Factorization
where L is a lower triangular matrix,
and U is an upper triangular matrix.
- Eigendecomposition
3. Vector Space
- Column space
- Nullspace
- Row space
- Left nullspace
- The dimension of column space
- The dimension of the nullspace
- The dimension of the row space
- The dimension of the left nullspace
4. Eigenvalues and Eigenvectors
