Math I - 2DE20

2DE20 Math I (Linear Algebra)

• Area: Mathematics
• Topics: Differential Equations · Linear Algebra
• Related courses: Calculus - 2WBB0 · Math II - 5EMA0 · Control Systems - 5ESD0

Recommended: 3b1b

Determinant

Triangular matrix: Product of diagonal
Transforms:

• Transpose: → Row operations = Col operations
• Row/Col operations:
• Interchange row/col:
• Multiply row/col by k:
• Inverse:
• Scale:

Invertible

• A is row equivalent to
• Ax=0 has the only solution x=0
• Ax=b has 1 unique solution
• != 0
• Column Space of A ==

Vector space

• Addition
  1. V is closed under addition
  2. Commutative
  3. Associative
  4. Identity element:
• Multiplication
  1. Distributive (scalar & vector)
  2. Identity element:

Subspace

A subspace of vector space V is a subset W of V that has:

1. The zero vector 0 of V is in W
2. is closed under vector addition
4. are trivial subspaces of V

Column Space

Span: set of linear combinations of vectors
Colume space: span of column vectors
Basis: minimal subset of span that spans the vector space

Theorem:

Matrix Rank

Eigenvalue, eigenvector

Topic: Linear Algebra

: characteristic polynomial

Eigenspace : containing all eigenvectors corresponding to

Orthogonality

• If is an orthogonal basis for subspace W in then:
• Spectral decomposition: Eigenvectors forming orthonormal basis of corresponding to eigenvalues :

Gram-Schmidt process

Convert arbitrary basis

Diagonalization

Similar matrix: A similar to B

: similar transformation → switching to eigenvectors basis → guarantee diagonal matrix
Same characteristic polynomial → same eigenvalues

Diagonalizing mxm matrix A into similar diagonal matrix D:

1. Find eigenvalues
2. Find eigenvectors

Theorem:

1. nxn matrix is diagonalizable if and only if it has n distinct eigenvectors
2. Eigenvectors corresponding to distinct eigenvalues are linearly independent
3. nxn matrix with n distinct eigenvalues is diagonalizable

Isomorphism: 1-1 linear transformation from vector space V onto vector space W
matrix: Matrix C for A relative to | -matrix for A

• : change of basis

Complex eigenvector, eigenvalue
Proof
Transform 2x2 matrix with complex eigen(value+vector) to rotation matrix:

1. Find eigenvalue
2. Find eigenvector

3. with:

Differential equation:

Topic: Differential Equations

Complex eigen(value+vector):
2x2
General solution:
Real solution:

Symmetric matrix:

Symmetric matrix:

Theorem:

1. Have n real eigenvalues
2. Eigenvectors corresponding to distinct eigenvalues are orthogonal
3. Eigenvectors form orthonormal basis Orthogonally diagonalizable:
4. Orthogonal symmetric, symmetric orthogonal

Orthogonal matrix: columns (rows) form orthonormal basis in