Bases and Coordinate Systems

A basis is a set of directions that lets you describe every vector in a space, with no unnecessary direction.

In the usual two-dimensional coordinate system, the basis vectors are:

e_1 = [1, 0], \quad e_2 = [0, 1]

So:

[3, 2] = 3e_1 + 2e_2

The coordinates 3 and 2 say how much of each basis direction to use.

Why Bases Matter

A basis gives coordinates their meaning.

If the basis changes, the coordinate numbers may change. The vector being described can still be the same geometric object.

This is one reason matrices are useful. A matrix can transform vectors from one coordinate description to another or move them into a new space.

This is also why bare coordinate numbers can mislead. The coordinate pair [3, 2] means "three of the first basis direction and two of the second." If those basis directions change, the same pair of numbers can describe a different geometric vector.

MATH-C04-T07-001Exercise: Read coordinates in a basis

Let e_1 = [1, 0] and e_2 = [0, 1].

Compute 4e_1 + 2e_2.

Check the vector

Compute it first, then check your number.

HintScale then add

Scale each basis vector, then add.

SolutionStandard basis

Coordinates are weights on basis directions. Scale each basis vector first:

4[1,0] + 2[0,1] = [4,0] + [0,2] = [4,2]

In the standard basis, the coordinates and the final vector have the same numbers.

MATH-C04-T07-004Exercise: Coordinates need basis vectors

Suppose a basis uses b_1 = [1, 0] and b_2 = [1, 1].

What vector is described by the coordinates [2, 3] in this basis?

Your answer

Compute it first, then check your number.

HintCoordinates are weights

Compute 2[1, 0] + 3[1, 1].

SolutionUse the frame

The coordinates [2, 3] mean 2 copies of b_1 plus 3 copies of b_2, not the standard vector [2, 3] by default:

2[1,0] + 3[1,1] = [2,0] + [3,3] = [5,3]

The basis frame changes how the coordinate list is read.

Not Every Set Is a Basis

In two dimensions, two directions form a basis only if they are not copies of the same direction.

For example, [1, 0] and [2, 0] do not form a basis for the plane. They only move along one line.

MATH-C04-T07-002Exercise: Spot a failed basis

Enter 1 if [1,0] and [2,0] form a basis for the full two-dimensional plane, or 0 if they do not.

Your answer

Compute it first, then check your number.

HintSame direction

Ask whether you can move upward using these two directions.

SolutionOnly one line

Enter 0. Both vectors lie on the horizontal axis. They can describe horizontal movement, but not every vector in the plane.

Basis As A Frame

Coordinates depend on the chosen basis.

With the standard basis:

[3,2] = 3[1,0] + 2[0,1]

But if the basis directions change, the coordinate numbers needed to describe the same geometric vector may also change.

MATH-C04-T07-003Exercise: Coordinates need a frame

Enter 1 if coordinate numbers only make full sense after the basis or frame is known.

Your answer

Compute it first, then check your number.

HintAsk relative to what

Coordinates say how much of each basis direction to use.

SolutionBasis gives meaning

Enter 1. A coordinate list describes amounts along chosen basis directions. Without the basis, the numbers do not fully say which geometric vector is being described.

Before Moving On

A basis is the measuring frame for a space. Coordinates only make sense relative to that frame.