Questions: The Standard Matrix of a Linear Transformation

Question 1

A student wants to find the standard matrix for T: ℝ² → ℝ² defined by T(x, y) = (2x + y, x − 3y). What is the correct procedure?

• A. Compute T(1, 0) = (2, 1) and T(0, 1) = (1, −3); assemble these as the columns of A
• B. Solve the system Ax = T(x) for A using row reduction on sample inputs
• C. Find vectors that A maps to e₁ and e₂ and use those as rows
• D. Average T's output over many random inputs to estimate A's entries

Answer: (A)

The standard matrix is assembled directly from the images of the basis vectors as columns — no equation solving required. T(e₁) = T(1, 0) = (2, 1) becomes column 1; T(e₂) = T(0, 1) = (1, −3) becomes column 2. This works because linearity guarantees T(x) = x₁·T(e₁) + x₂·T(e₂), which is exactly what Ax computes.

Question 2

The standard matrix of a 90° counterclockwise rotation is [[0, −1], [1, 0]]. Why is the first column [0, 1]ᵀ rather than [1, 0]ᵀ?

• A. Because e₁ = [1, 0]ᵀ maps to [0, 1]ᵀ under a 90° counterclockwise rotation, and columns encode where basis vectors go
• B. Because [1, 0]ᵀ is reserved for identity matrices and cannot appear as a rotation column
• C. Because columns of a rotation matrix must always be perpendicular to the rows
• D. Because rows encode where basis vectors go; the first row therefore cannot match e₁

Answer: (A)

Column j of the standard matrix is T(eⱼ). Rotating e₁ = [1, 0]ᵀ by 90° counterclockwise produces [0, 1]ᵀ — that becomes column 1. Rotating e₂ = [0, 1]ᵀ produces [−1, 0]ᵀ — that becomes column 2. The common mistake in option D (rows encode basis images) would produce the transpose, which represents a different transformation.

Question 3

The standard matrix of a linear transformation is independent of what basis you use for the input and output spaces.

• T. True
• F. False

Answer: False

The 'standard' qualifier specifically means the standard basis is used for both domain and codomain — that's what makes it standard. Use a different basis and the same transformation T has a different matrix representation. The study of change of basis is precisely about converting between these representations. Basis-independence would mean every transformation has only one matrix, which is false.

Question 4

To construct the standard matrix of T: ℝⁿ → ℝᵐ, you should evaluate T on most vector in ℝⁿ.

• T. True
• F. False

Answer: False

You only need T evaluated on the n standard basis vectors e₁, …, eₙ. By linearity, any vector x = x₁e₁ + ··· + xₙeₙ gives T(x) = x₁T(e₁) + ··· + xₙT(eₙ), which the matrix product Ax reproduces automatically. The entire transformation on an infinite domain is encoded in just n basis images — this is the power of linearity.

Question 5

Why is it sufficient to know T(e₁), T(e₂), …, T(eₙ) to completely determine a linear transformation T: ℝⁿ → ℝᵐ?

Model answer: Any vector x ∈ ℝⁿ decomposes as x = x₁e₁ + ··· + xₙeₙ. Linearity then gives T(x) = x₁T(e₁) + ··· + xₙT(eₙ). Knowing T on the basis vectors determines T on every linear combination of them — which is every vector in ℝⁿ.

This is the fundamental theorem behind the standard matrix. A linear transformation is completely and uniquely determined by its values on a basis, because linearity 'extends' those values to all of ℝⁿ without ambiguity. The matrix is just the data structure that packages the n basis images so that Ax replicates the extension automatically.