How are coordinates defined for a vector $u\in U\subseteq\mathbb{R}^N$?

If $\mathcal{B}=(v_1,\dots,v_k)$ is a basis of $U$, then $u$ can be written as $u=\lambda_1 v_1+\cdots+\lambda_k v_k$. The numbers $\lambda_1,\dots,\lambda_k$ are the coordinates of $u$ with respect to $\mathcal{B}$. There are exactly $k$ coordinates because there are $k$ basis vectors.

Linear Algebra 25 | Coordinates with respect to a Basis [dark version]

Q: What exactly makes a set of vectors a basis for a subspace?

A basis for a subspace $U$ must satisfy two conditions: (1) it spans $U$, meaning every $u\in U$ can be expressed as a linear combination of the basis vectors; and (2) it is linearly independent, which ensures the representation is unique. Both ingredients are required to make coordinates well-defined.

Q: Why are the coordinates unique?

Uniqueness comes from linear independence. If $u$ could be written in two different ways using the same basis vectors, subtracting the two expressions would produce a nontrivial linear combination of basis vectors equaling zero—contradicting linear independence. So each $u$ has exactly one coordinate list relative to $\mathcal{B}$.

Q: What notational confusion can happen when coordinates are written as a column vector?

A coordinate column vector has $k$ entries, while the original vector $u$ lives in $\mathbb{R}^N$. Writing an equality between them can be misleading. The correct interpretation is that the coordinate column vector is shorthand for the linear combination $u=\lambda_1 v_1+\cdots+\lambda_k v_k$. Indexing the notation by the basis $\mathcal{B}$ helps keep the meaning clear.

TL;DR

A basis of a subspace must both span the subspace and be linearly independent.

Briefing Cornell Notes

Briefing

Coordinates with respect to a basis turn one and the same vector into different coordinate lists—depending on which spanning, linearly independent vectors are chosen as the reference. That flexibility matters because a “better” basis can make calculations dramatically simpler, even though the underlying vector in the subspace never changes.

The discussion starts with the standard basis in $R^{2}$ : the unit vectors $e_{1}$ and $e_{2}$ form an orthogonal grid aligned with the axes. Any vector $v$ in the plane can be described by how far to move in the $x$ -direction and $y$ -direction, including negative moves. Those two movement amounts are exactly the coordinates of $v$ relative to the standard basis.

Switching to a different basis changes the coordinate system. In $R^{2}$ , choosing two new independent vectors as basis elements creates a new grid that need not be right-angled. The same geometric vector $v$ can then be reached by moving some number of steps along the first basis vector and some number along the second. In the example, the coordinates become $(1, 1)$ instead of the standard-basis coordinates, illustrating why basis choice can streamline computation.

The core definition is then generalized to a subspace $U \subseteq R^{N}$ . If $B = (v_{1}, \dots, v_{k})$ is a basis of $U$ , then every vector $u \in U$ can be written as a linear combination $u = λ_{1} v_{1} + λ_{2} v_{2} + \dots + λ_{k} v_{k} .$ The coefficients $λ_{1}, \dots, λ_{k}$ are called the coordinates of $u$ with respect to $B$ . Linear independence guarantees uniqueness: there is only one way to express $u$ using those basis vectors, so the coordinates are well-defined. Because there are $k$ basis vectors, the coordinate list always has length $k$ , not $N$ .

To connect this to practical calculations, the transcript warns about notation: writing the coordinates as a column vector is a shorthand for the linear combination, not a literal equality between vectors with different dimensions. A common way to reduce confusion is to index the notation by the basis $B$ .

Finally, two $R^{3}$ examples show how coordinates arise from solving for the linear combination. For a non-standard basis, the vector $(1, 2, - 1)$ is expressed by reading off how many times each basis vector must be used, yielding coordinates $(1, 2, 1)$ . Another vector $(3, 0, 0)$ ends up requiring only the first basis vector with coefficient $- 1$ , giving coordinates $(- 1, 0, 0)$ . The takeaway is straightforward: coordinates are basis-dependent, but the basis-dependent coefficients are uniquely determined and can make solving for representations far easier.

Cornell Notes

A basis of a subspace $U \subseteq R^{N}$ is a linearly independent set that spans $U$ . For a basis $B = (v_{1}, \dots, v_{k})$ , every vector $u \in U$ can be written uniquely as $u = λ_{1} v_{1} + \dots + λ_{k} v_{k}$ . The scalars $λ_{1}, \dots, λ_{k}$ are the coordinates of $u$ with respect to $B$ . Changing the basis changes the coordinates because it changes the grid used to describe vectors, even though the geometric vector itself stays the same. Choosing a more convenient basis can make the coefficients—and thus computations—much simpler.

Why do the same vector have different coordinate lists in $R^{2}$ ?

Coordinates depend on the chosen basis. With the standard basis

e_{1}, e_{2}

, coordinates record how far to move along the

x

- and

y

-axes. If a different pair of independent vectors is chosen as the basis, the plane gets a new grid aligned with those vectors, so the same geometric vector is reached by different step counts along the new directions. That’s why the coordinate pair changes even though the vector in the plane does not.

What exactly makes a set of vectors a basis for a subspace?

A basis for a subspace

U

must satisfy two conditions: (1) it spans

U

, meaning every

u \in U

can be expressed as a linear combination of the basis vectors; and (2) it is linearly independent, which ensures the representation is unique. Both ingredients are required to make coordinates well-defined.

How are coordinates defined for a vector $u \in U \subseteq R^{N}$ ?

B = (v_{1}, \dots, v_{k})

is a basis of

U

, then

u

can be written as

u = λ_{1} v_{1} + \dots + λ_{k} v_{k}

. The numbers

λ_{1}, \dots, λ_{k}

are the coordinates of

u

with respect to

B

. There are exactly

k

coordinates because there are

k

basis vectors.

Why are the coordinates unique?

Uniqueness comes from linear independence. If

u

could be written in two different ways using the same basis vectors, subtracting the two expressions would produce a nontrivial linear combination of basis vectors equaling zero—contradicting linear independence. So each

u

has exactly one coordinate list relative to

B

What notational confusion can happen when coordinates are written as a column vector?

A coordinate column vector has

k

entries, while the original vector

u

lives in

R^{N}

. Writing an equality between them can be misleading. The correct interpretation is that the coordinate column vector is shorthand for the linear combination

u = λ_{1} v_{1} + \dots + λ_{k} v_{k}

. Indexing the notation by the basis

B

helps keep the meaning clear.

How do the $R^{3}$ examples produce coordinates for a non-standard basis?

The method is to express the target vector as a linear combination of the basis vectors. In the example with target

(1, 2, - 1)

, the coefficients are determined by matching components via the basis vectors, resulting in coordinates

(1, 2, 1)

. For

(3, 0, 0)

, the representation uses only the first basis vector with coefficient

- 1

, giving coordinates

(- 1, 0, 0)

Review Questions

Given a basis $B = (v_{1}, v_{2}, v_{3})$ for a subspace $U$ , what guarantees that the coordinates of a vector $u \in U$ are unique?
If a vector $u$ has coordinates $(λ_{1}, λ_{2})$ with respect to one basis in $R^{2}$ , what changes when switching to a different basis?
Why is it misleading to interpret a coordinate column vector as the same object as the original vector in $R^{N}$ ?

Key Points

1
A basis of a subspace must both span the subspace and be linearly independent.
2
Coordinates of a vector are the coefficients in its unique linear combination of basis vectors.
3
Changing the basis changes the coordinate list because it changes the grid used to describe vectors.
4
A basis of size $k$ produces coordinate vectors with exactly $k$ entries, even if the ambient space has dimension $N$ .
5
Uniqueness of coordinates is guaranteed by linear independence.
6
Writing coordinates as a column vector is shorthand for a linear combination, not a literal equality of vectors in different dimensions.

Highlights

In R2, switching from the standard basis to an oblique basis changes the coordinates of the same geometric vector because the underlying grid rotates and skews.

For U⊆RN with basis B=(v1​,…,vk​), every u∈U has a unique coordinate representation u=∑j=1k​λj​vj​.

Coordinate notation can be confusing: a k-entry coordinate column vector represents coefficients, not a vector in RN.

In the R3 examples, coordinates are found by expressing the target vector as a linear combination of the chosen non-standard basis vectors, yielding results like (1,2,1) and (−1,0,0).

Topics

Basis and Coordinates
Subspace Representation
Linear Independence
Coordinate Notation
Examples in R3