MTH2021 · Linear Algebra with Applications
Coordinates, Change of Basis & Direct Sums
Week 5 turns a basis into a coordinate system: relative to a basis B, every vector has a unique coordinate vector [v]_B, and the change-of-basis matrix P converts coordinates between two bases via [v]_{B'} = P[v]_B. It also introduces the direct sum V ⊕ W. These computations appear in Quiz 5 and Test 2 (Weeks 3–7), and change-of-basis reasoning underpins the matrix-of-a-map and diagonalisation chapters that follow.
What this chapter covers
- 01Unique expansion of v in a basis B; the coordinate vector [v]_B ∈ ℝⁿ
- 02Coordinates in polynomial and matrix spaces, e.g. [a₀ + a₁x + a₂x²] in {1, x, x²}
- 03Change-of-basis matrix P_{B'←B} whose columns are the old basis vectors' coordinates in the new basis
- 04The transformation law [v]_{B'} = P_{B'←B} [v]_B
- 05P_{B'←B} is invertible and (P_{B'←B})⁻¹ = P_{B←B'}
- 06Sum of subspaces V + W = {v + w}; it is a direct sum V ⊕ W when V ∩ W = {0}
- 07Uniqueness: the sum is direct ⇔ every element writes uniquely as v + w
- 08Dimension of a direct sum: dim(V ⊕ W) = dim(V) + dim(W)
Coordinate vector and change of basis in ℝ²
- +1(a) Solve a(1,1) + b(1,−1) = (3,1). Componentwise: a + b = 3 and a − b = 1. Adding gives 2a = 4 ⇒ a = 2; then b = 3 − 2 = 1.
- +1(a) So [v]_B = (2, 1): v = 2·(1,1) + 1·(1,−1). Check: 2(1,1) + (1,−1) = (2+1, 2−1) = (3, 1) ✓.
- +1(b) The change-of-basis matrix from B to the standard basis has the B-vectors as its columns: P_{E←B} = [[1, 1],[1, −1]] (column 1 = (1,1), column 2 = (1,−1)).
- +1(b) Recover v from its B-coordinates: v = P_{E←B} [v]_B = [[1,1],[1,−1]]·(2,1)ᵀ = (1·2 + 1·1, 1·2 + (−1)·1)ᵀ = (3, 1)ᵀ ✓. (Going the other way uses (P_{E←B})⁻¹ = P_{B←E}.)
Key terms
- Coordinate vector [v]_B
- The unique tuple (c₁, …, cₙ) of scalars with v = c₁v₁ + … + cₙvₙ for a basis B = {v₁, …, vₙ}; it lives in ℝⁿ and identifies V with ℝⁿ.
- Change-of-basis matrix P_{B'←B}
- The matrix whose columns are the coordinate vectors of the old basis B written in the new basis B'; it satisfies [v]_{B'} = P_{B'←B}[v]_B.
- Transformation law
- [v]_{B'} = P_{B'←B}[v]_B: multiplying a vector's B-coordinates by the change-of-basis matrix gives its B'-coordinates.
- Inverse change of basis
- P_{B'←B} is invertible and (P_{B'←B})⁻¹ = P_{B←B'}, so you can always convert coordinates back the other way.
- Sum of subspaces
- V + W = {v + w : v ∈ V, w ∈ W}, the smallest subspace containing both V and W.
- Direct sum V ⊕ W
- A sum in which V ∩ W = {0}; equivalently every element of V + W writes uniquely as v + w, and dim(V ⊕ W) = dim(V) + dim(W).
Coordinates, Change of Basis & Direct Sums FAQ
What is a coordinate vector, really?
Once you fix a basis B = {v₁, …, vₙ}, every vector v has one and only one expansion v = c₁v₁ + … + cₙvₙ, and the coefficients (c₁, …, cₙ) are its coordinate vector [v]_B. This turns abstract vectors — polynomials, matrices, anything — into ordinary columns in ℝⁿ, which is why the whole computational machinery transfers.
How do I build a change-of-basis matrix?
The columns of P_{B'←B} are the coordinate vectors, in the new basis B', of the old basis vectors. When the new basis is the standard basis, those coordinates are just the old basis vectors themselves, so P is simply the basis vectors written as columns. Then [v]_{B'} = P[v]_B.
Which way does the change-of-basis matrix go?
Read the arrow: P_{B'←B} takes B-coordinates and returns B'-coordinates, so it multiplies [v]_B. To reverse direction use its inverse, since (P_{B'←B})⁻¹ = P_{B←B'}. Mixing up the direction is the most common slip, so always write the subscripts.
When is a sum of subspaces a direct sum?
When the two subspaces meet only at 0 (V ∩ W = {0}). That is equivalent to every vector in V + W having a unique decomposition v + w, and it gives the clean dimension formula dim(V ⊕ W) = dim(V) + dim(W). Direct sums reappear in the Fundamental Theorem of Linear Algebra (ℝⁿ = row(A) ⊕ ker(A)).
Where is this assessed?
Coordinate vectors and change of basis are examined in Quiz 5 and Test 2 (Weeks 3–7). They also underpin the matrix-of-a-linear-map and diagonalisation chapters, so getting the P notation right now pays off directly in the final.
Exam move
Practise the two-step drill: solve a small linear system to get [v]_B, and build P by stacking basis vectors (or their coordinates) as columns. Always write the arrow subscripts P_{B'←B} so you multiply on the correct side, and rehearse inverting P to go the other way. Coordinates and change of basis are examined in Quiz 5 and Test 2 (Weeks 3–7) and set up diagonalisation later. When the direction confuses you, ask Sia to trace which coordinates go in and which come out.
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