# Section3.5Bases as Coordinate Systems¶ permalink

##### Objectives
1. Learn to view a basis as a coordinate system on a subspace.
2. Recipes: compute the -coordinates of a vector, compute the usual coordinates of a vector from its -coordinates.
3. Picture: the -coordinates of a vector using its location on a nonstandard coordinate grid.
4. Vocabulary: -coordinates.

In this section, we interpret a basis of a subspace as a coordinate system on and we learn how to write a vector in in that coordinate system.

##### Example

Consider the standard basis of from this example in Section 3.4:

According to the above fact, every vector in can be written as a linear combination of with unique coefficients. For example,

In this case, the coordinates of are exactly the coefficients of

What exactly are coordinates, anyway? One way to think of coordinates is that they give directions for how to get to a certain point from the origin. In the above example, the linear combination can be thought of as the following list of instructions: start at the origin, travel units north, then travel units east, then units down.

##### Definition

Let be a basis of a subspace and let

be a vector in The coefficients are the coordinates of with respect to . The -coordinate vector of is the vector

If we change the basis, then we can still give instructions for how to get to the point but the instructions will be different. Say for example we take the basis

We can write in this basis as In other words: start at the origin, travel northeast times as far as then units east, then units down. In this situation, we can say that is the -coordinate of is the -coordinate of and is the -coordinate of

The above definition gives a way of using to label the points of a subspace of dimension a point is simply labeled by its -coordinate vector. For instance, if we choose a basis for a plane, we can label the points of that plane with the points of

##### Example

Let

These form a basis for a plane in We indicate the coordinate system defined by by drawing lines parallel to the -axis” and -axis”:

We can see from the picture that the -coordinate of is equal to as is the -coordinate, so Similarly, we have

##### Recipes: B -coordinates

If is a basis for a subspace and is in then

Finding the -coordinates of means solving the vector equation

in the unknowns This generally means row reducing the augmented matrix