Standard Basis

Standard Basis

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

Linear Combination

The components of the vector that form the basis are:

Standard Basis

Examples

Standard Basis Example

Standard Basis Example


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

Example

Which pairs of the following vectors form a basis?

Vector Basis Example

Vector Basis Example

Vector Basis Example

Vector Basis Example

Orthogonal Basis

Orthogonal Basis

 

The two basis vectors are mutually perpendicular.

Orthonormal Basis

Orthonormal Basis


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

Basis Vector

Basis Vectors

Basis Vectors

Standard Basis

The base formed by Vector and Vector 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 Vector = (2, 1), Vector = (1, 4) and Vector = (5, 6).

1. Determine if Vector and Vector form a basis.

Standard Basis Example

2. Express Vector as a linear combination.

Linear Combination Example

Linear Combination Example

Linear Combination Solution

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

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


A vector Vector has coordinates (3, 5) in the standard basis. Which coordinates of Vector will be referred to the basis of Vector = (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 Vector in the basis B are (7/3, 1/3).