Linear Algebra 25 | Coordinates with respect to a Basis

TL;DR

A basis defines the coordinate “grid,” so the same vector can have different coordinate lists under different bases.

Briefing Cornell Notes

Briefing

Coordinates with respect to a basis turn the same vector into different coordinate lists—often making calculations easier—because a basis defines the “grid” used to describe vectors. In the familiar setting of R2, the standard basis uses the usual x- and y-directions, so a vector is described by how far to move right/left and up/down. Switching to a different basis replaces that orthogonal grid with a new one determined by two chosen basis vectors, which may be skewed or non-right-angled. The same geometric vector can then have simpler coordinates in the new system; for instance, a vector that needs fractional steps in the standard basis might become (1,1) when expressed using the new basis vectors.

This idea generalizes to any subspace U of RN. If B = {v1, v2, …, vk} is a basis of U, then every vector u in U can be written as a linear combination u = λ1 v1 + λ2 v2 + … + λk vk. The coefficients (λ1, …, λk) are defined as the coordinates of u with respect to the basis B. A key point is uniqueness: because the basis vectors are linearly independent, there is only one way to represent u as such a linear combination. That uniqueness guarantees the coordinates are well-defined, and there are exactly k coordinate numbers—one for each basis vector.

Once the coordinates are known, the original vector can be reconstructed from them, which motivates a compact notation: the coordinates are often stacked into a column vector. However, this column vector does not mean the original vector u literally has n components; it only encodes the coefficients relative to the basis. To avoid confusion, the basis is typically included in the notation (so the reader remembers the coordinates depend on B). In computations inside the subspace, working with these coordinate lists can streamline algebra, though the transcript notes that handling changes between different bases is a later topic.

Two worked examples make the definition concrete. In R3, a non-standard basis is used to find coordinates of u = (1, 2, −1). By matching components through the linear combination—using the third basis vector once to produce −1, the second twice to produce 2, and the first once to complete the combination—the coordinates come out as (1, 2, 1) relative to that basis. A second example takes u~ = (3, 0, 0), which is achieved by taking −1 times the first basis vector, yielding coordinates (−1, 0, 0). The takeaway is straightforward: “coordinates” are not intrinsic to the vector alone; they are the coefficients that express the vector in terms of a chosen basis, and choosing a better basis can make those coefficients easier to compute.

Cornell Notes

A basis of a subspace U provides a unique way to write every vector u in U as a linear combination of basis vectors. If B = {v1, …, vk} is a basis, then u = λ1 v1 + … + λk vk, and the numbers (λ1, …, λk) are the coordinates of u with respect to B. Linear independence guarantees these coordinates are uniquely determined, so the coordinate list is well-defined. Coordinates can be arranged into a column vector for convenience, but that column vector represents coefficients relative to B—not the original vector’s n-dimensional components. Because different bases define different “grids,” the same vector can have different coordinate lists under different bases, sometimes with simpler numbers.

Why do the same vector have different coordinates after changing the basis?

A basis determines the directions used to build the vector. In R2, the standard basis corresponds to the usual x- and y-directions, so coordinates measure steps along those axes. Choosing a different pair of basis vectors creates a different grid (possibly skewed), so the same geometric vector is expressed using different linear combinations, producing different coordinate lists.

What exactly are “coordinates of u with respect to a basis B”?

If B = {v1, …, vk} is a basis of a subspace U ⊆ RN, then every u ∈ U can be written as u = λ1 v1 + … + λk vk. The coordinates of u with respect to B are the coefficients (λ1, …, λk). These coefficients exist for every u in U because the basis spans U.

How does linear independence affect the coordinate values?

Linear independence makes the representation unique. If the basis vectors were not independent, a vector could be written as different linear combinations, leading to multiple possible coordinate lists. With a true basis, there is exactly one set of coefficients (λ1, …, λk) for each u.

Why is it potentially confusing to treat the coordinate column vector as if it were the original vector?

The coordinate column vector has k entries (the number of basis vectors), while the original vector u lives in RN and has n components. The column vector stores coefficients relative to B, and the equality should be read as “the coefficients multiply the basis vectors to produce u,” not as “u literally equals a k-component vector.”

How were coordinates found in the R3 example using a non-standard basis?

For u = (1, 2, −1), the method was to choose coefficients that reproduce each component through the linear combination of the basis vectors. The transcript describes starting with the third coordinate: using the third basis vector once to get −1, using the second basis vector twice to get 2, and then using the first basis vector once to complete the combination. The resulting coordinates relative to that basis are (1, 2, 1).

What does the second R3 example show about expressing a vector using basis vectors?

For u~ = (3, 0, 0), the vector is obtained by taking −1 times the first basis vector, with no contribution from the other basis vectors. That yields coordinates (−1, 0, 0) relative to the chosen non-standard basis, illustrating that coordinates can be simple even when the basis is not.

Review Questions

Given a basis B = {v1, v2, …, vk} of a subspace U, what conditions ensure that the coordinates of any u ∈ U exist and are unique?
If two different bases are used for the same vector u, what changes and what stays the same?
When writing coordinates as a column vector, how should the equality involving u be interpreted to avoid confusion?

Key Points

1
A basis defines the coordinate “grid,” so the same vector can have different coordinate lists under different bases.
2
If B = {v1, …, vk} is a basis of U, every u ∈ U can be written uniquely as u = λ1 v1 + … + λk vk.
3
The coordinates of u with respect to B are exactly the coefficients (λ1, …, λk) in that linear combination.
4
Uniqueness of coordinates comes from linear independence of the basis vectors.
5
Coordinate column vectors store coefficients relative to the basis and should not be mistaken for the original n-component vector.
6
Knowing coordinates allows reconstruction of the original vector by combining the basis vectors with those coefficients.
7
Choosing a more suitable basis can make coordinate values—and therefore calculations—simpler.

Highlights

Changing the basis changes the grid used to describe vectors, so coordinates for the same vector can become simpler or more complicated.

A basis of a subspace guarantees every vector in the subspace can be expressed as a linear combination of the basis vectors.

Linear independence is what makes the coordinate list unique for each vector.

Coordinate notation as a column vector represents coefficients relative to the basis, not the vector’s native component structure.

In the R3 example, matching component contributions through the basis vectors produced coordinates (1, 2, 1) for (1, 2, −1) under a non-standard basis.

Topics

Basis of Subspace
Coordinates
Linear Independence
Linear Combinations
Non-Standard Bases