02. Linear Transformation

Linear Transformation

A linear transformation is a mapping from one vector space to another that preserves the vector space conditions.

Matrix

A matrix is a rectangular grid of cells. Every linear transformation has a matrix representation.

Rank

Use the elimination algorithm of Gauß to simplify the matrix until everything is in the top left corner, all the elements in one triangular half are 0.

Then count how many linearly independent vectors there are left. This is the rank.

Solving of linear equations

Put the coefficients into a matrix and the unknown variables into a vector. Multiply and try to make equal to the right side.

Solving of equations with matrix variables in them
High-level algebra with matrices

Let A, B be matrices. Let I be the unit matrix. Let λ be a scalar.

  • A⋅{\bf v}={\bf b}: Use elimination algorithm by Gauß to find v.
  • A⋅{\bf x}=λ⋅{\bf x}: Introduce I to find the identical equivalent equation A⋅{\bf x}=λ⋅I⋅{\bf x}. Use the cross product associativity to get (A-λ⋅I)⋅{\bf x}=0. First, get λ by solving \det(A-λ⋅I)=0 for λ. Then use the elimination algorithm by Gauß to solve for x.
Without inhomogenity
With inhomogenity

Symmetry

Antisymmetric Matrix (skewed symmetry)

A antisymmetric matrix is a matrix A where:

A^T=-A

For a - the entries of A, this means:

a_{ij}=-a_{ji} \forall i,j\in \{1,\ldots,n\}
Cross Product

The cross product can be written as a matrix multiplication:

{\bf a}⨯{\bf b}=S_a⋅{\bf b}

Where S_a is:

S_a:=\begin{pmatrix} 0 & -a_3 & a_2\\ a_3 & 0 & -a_1 \\ -a_2 & a_1 & 0 \end{pmatrix}

Determinant

See Matrix Properties: Determinant.

Differential equations

Eigenvalue

See Matrix Properties: Eigenvalue.

Author: Danny (remove the ".nospam" to send)

Last modification on: Sat, 04 May 2024 .