Standard Basis

Two linearly independent vectors, and form a basis, because any vector in the plane can be set as a linear combination of them.

The components of the vector that form the basis are:

Examples

The two vectors that form a basis cannot be parallel to one another.

Example

Which pairs of the following vectors form a basis?

Orthogonal Basis

The two basis vectors are mutually perpendicular.

Orthonormal Basis

The two basis vectors are mutually perpendicular and their magnitude is one.

Standard Basis

The base formed by and is called the standard basis or canonical basis.

The standard basis is the base that is commonly used, so if nothing is noticed, it should be working on that basis.

Examples

Given the vectors = (2, 1), = (1, 4) and = (5, 6).

1. Determine if and form a basis.

2. Express as a linear combination.

3. Calculate the coordinates of with respect to the base.

The coordinates of with respect to the base are: (2, 1).

A vector has coordinates (3, 5) in the standard basis. Which coordinates of will be referred to the basis of = (1, 2), = (2, 1)?

(3, 5) = a (1, 2) + b (2, 1)

3 = a + 2b a = 3 − 2b a = 7/3

5 = 2a + b 5 = 2 (3 − 2b) + b b = 1/3

The coordinates of in the basis B are (7/3, 1/3).