# 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

### 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).