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🎓 University Year 1 · Lesson 10 of 15

Basis Changer: Coordinates Are a Choice

A vector is an arrow in space; its list of numbers is only how that arrow looks against a particular set of measuring axes — change the axes and the numbers change, though the arrow does not.

University Year 1Calculus II / Linear Algebra
Basis Changer: Coordinates Are a Choice — illustration
💡
The big idea: We usually write a vector as a column of numbers, but those numbers are not the vector itself — they are its coordinates relative to a chosen basis. Pick a different basis and the very same arrow gets a different list of numbers. A change-of-basis matrix is the dictionary that translates coordinates from one basis to another. Choosing the right basis is one of the most powerful moves in mathematics: it can turn an ugly transformation into a simple one.
🎯 By the end, you'll be able to
  • Explain the difference between a vector and its coordinates in a basis
  • Build the change-of-basis matrix whose columns are the new basis vectors
  • Convert a vector's coordinates between the standard basis and another basis
  • Use the inverse matrix to go the other direction
  • Recognise why a well-chosen basis simplifies a linear transformation
📎 You should already know
  • Vectors and linear combinations
  • Matrix–vector multiplication
  • Inverse of a 2×2 matrix

The numbers are not the vector

When you write a vector as (2, 4), you are really giving an instruction: “go 2 steps along the first axis and 4 along the second.” That instruction only makes sense once you have agreed what the axes are. The default choice is the standard basis — the unit arrows e1 = (1, 0) and e2 = (0, 1).

But nothing forces that choice. Any two arrows that are not parallel can serve as axes. Against a different pair, the same point in the plane needs a different pair of numbers to describe it.

🔑 A basis is a coordinate system
A basis of the plane is a pair of vectors that are linearly independent (not parallel). Every vector can be written as exactly one linear combination of them, and those two weights are the vector's coordinates in that basis. Change the basis and you change the coordinates — but the underlying arrow stays put.
\[ \vec{v} = c_1\,\vec{b}_1 + c_2\,\vec{b}_2 \quad\Longrightarrow\quad [\vec{v}]_B = \begin{bmatrix} c_1 \\ c_2 \end{bmatrix} \]
The coordinates of v in the basis B = {b₁, b₂} are the weights c₁, c₂ that rebuild v from the basis vectors.
🎮 Basis Changer LIVE
See the same vector's coordinates change when you change the basis.

The change-of-basis matrix

Suppose you know a vector by its coordinates in a basis B and want its ordinary standard coordinates. Just build the matrix P whose columns are the basis vectors b1, b2 written in standard coordinates. Then multiplying by P rebuilds the vector.

To go the other way — from standard coordinates back to B-coordinates — multiply by the inverse, P−1.

\[ [\vec{v}]_{\text{std}} = P\,[\vec{v}]_B, \qquad [\vec{v}]_B = P^{-1}\,[\vec{v}]_{\text{std}} \]
P has the new basis vectors as its columns. It converts B-coordinates into standard ones; its inverse does the reverse.
📝 Worked example: Let B have basis vectors b₁ = (1, 1) and b₂ = (−1, 1). The vector v has standard coordinates (2, 4). Find its coordinates in B.
  1. Write the change-of-basis matrix P = [[1, −1], [1, 1]] — the basis vectors as columns.
  2. We need [v]_B with P[v]_B = (2, 4). Solve the system a − b = 2 and a + b = 4.
  3. Add the two equations: 2a = 6, so a = 3. Then b = 4 − a = 1.
  4. Check by rebuilding: 3·(1,1) + 1·(−1,1) = (3 − 1, 3 + 1) = (2, 4). ✓
✓ In basis B the coordinates are <strong>(3,&nbsp;1)</strong>: the same arrow, described against the tilted axes.
📝 Worked example: Using the inverse, confirm [v]_B for the same v = (2, 4).
  1. The determinant of P = [[1, −1], [1, 1]] is (1)(1) − (−1)(1) = 2.
  2. So \( P^{-1} = \tfrac{1}{2}\begin{bmatrix} 1 & 1 \\ -1 & 1 \end{bmatrix} \).
  3. Multiply: \( P^{-1}(2,4) = \tfrac{1}{2}(2+4,\; -2+4) = \tfrac{1}{2}(6, 2) \).
✓ This gives <strong>(3,&nbsp;1)</strong>, matching the direct solution.
✨ Columns of P are the new axes
The easy direction is always B-coordinates → standard, because the columns of P literally are the new basis vectors. So the matrix that takes coordinates out of a basis is the simple one to write; getting coordinates into a basis needs its inverse.

Why bother changing basis?

A matrix that mixes both coordinates together can look complicated. But choose the eigenvectors as your basis and the same transformation becomes a plain diagonal scaling — each axis stretched by its own eigenvalue, with no mixing. Fourier analysis, principal component analysis and the normal modes of a vibrating system are all just the art of finding the basis in which a hard problem becomes easy.

⚠️ A basis must be independent
The two vectors you pick must not be parallel. If they are — say (1, 2) and (2, 4) — then P has determinant zero, it has no inverse, and some vectors get no coordinates while others get infinitely many. Independence is exactly what ensures every vector has one and only one coordinate list.

Check your understanding

1. What are the coordinates of a vector in a basis B = {b₁, b₂}?
Coordinates in B are the unique weights that rebuild the vector as a combination of the basis vectors.
2. You know a vector's coordinates in basis B. How do you get its standard coordinates?
P has the basis vectors as columns, so [v]_std = P[v]_B rebuilds the vector in standard coordinates.
3. With b₁ = (1, 1) and b₂ = (−1, 1), the vector 2b₁ + 3b₂ has which standard coordinates?
2·(1,1) + 3·(−1,1) = (2 − 3, 2 + 3) = (−1, 5).
4. Why can't {(1, 2), (2, 4)} be used as a basis of the plane?
(2,4) = 2·(1,2), so the vectors are parallel; they cannot describe every point uniquely and det P = 0.
5. What is the main payoff of switching to an eigenvector basis?
In the eigenvector basis the matrix is diagonal: each axis is simply scaled by its eigenvalue, which is far easier to work with.
✅ Key takeaways
  • A vector is a fixed arrow; its coordinates are just how it looks against a chosen basis, and a different basis gives different numbers.
  • The change-of-basis matrix P has the new basis vectors as its columns and converts B-coordinates to standard: [v]_std = P[v]_B.
  • To go from standard coordinates into the basis, multiply by the inverse: [v]_B = P⁻¹[v]_std.
  • A basis must be linearly independent, or P is singular and coordinates fail to be unique.
  • Choosing a clever basis — often the eigenvectors — can turn a complicated transformation into a simple diagonal scaling.