Linear Algebra cheatsheet

Operations with Vectors

A matrix with a single column (column vector) or a single row (row vector).

A vector inis represented as:

Scalar

A real number used to multiply a vector.

Sum of Vectors

Ifand, the sum is:

Subtraction of Vectors

Ifand, the subtraction is:

Product by a Scalar

Ifandis a scalar, the product is:

Dot Product

Ifand, the dot product is:

Cross Product (only in)

Ifand, the cross product is:

Norm or Magnitude of a Vector

The norm of a vectoris:

Projection of a Vector

The projection of a vectoronto a vectoris defined as:

Angle Between Vectors

The anglebetween two vectorsandis given by:

Matrices

Matrix

A rectangular collection of numbers arranged in rows and columns.

Dimensions

A matrix ofhasrows andcolumns.

Square Matrix

A matrix is square if.

Transpose of a Matrix

Ifis a matrix of, its transposeis the matrix ofobtained by swapping rows and columns:

Symmetric Matrix

A symmetric matrix is a square matrix that is equal to its transpose. That is, a matrixis symmetric if:

This implies thatfor alland.

Symmetric matrices frequently appear in linear algebra and optimization problems.

Example of a symmetric matrix:

Identity Matrix

The identity matrixis a square matrix of sizein which the elements of the main diagonal areand the rest of the elements are.

• This matrix plays a role similar to the numberin the multiplication of numbers.
• andfor any compatible matrix.

Diagonal Matrix

A diagonal matrix is a square matrix in which all elements outside the main diagonal are. The elements of the diagonal can be any number.

• The product of diagonal matrices is commutative:.
• The determinant of a diagonal matrix is the product of the diagonal elements:;

Operations with Matrices

Sum of Matrices

Ifandare matrices of, the sum is:

That is,

Similarly

Product by a Scalar

Ifis a matrix ofandis a scalar:

That is,

Product of Matrices

Ifis a matrix ofandis a matrix of, the product is:

That is,

Properties of Matrix Product

• Associative:
• Distributive:
• Not Commutative: In general,

Determinant of a Matrix

Determinant of aMatrix

The determinant of a matrixofis:

Determinant of aMatrix

For a matrix:

Sarrus’ Rule (for):

Properties of the Determinant

• (whereis the size of the matrix)

Inverse of Matrices

Inverse Matrix

Ifis a square matrix of, its inverseis the matrix such that:

whereis the identity matrix of size.

Matrix

Adjoint Matrix Method

whereis the transpose of the cofactor matrix.