Questions: Matrix Representation of Linear Transformations

Question 1

A linear transformation T: R² → R² sends e₁ = (1, 0) to (3, −1) and e₂ = (0, 1) to (2, 5). What is the matrix A representing T in the standard basis?

• A. [[3, 2], [−1, 5]] — first column is T(e₁), second column is T(e₂)
• B. [[3, −1], [2, 5]] — first row is T(e₁), second row is T(e₂)
• C. [[−1, 5], [3, 2]] — columns are ordered by the output components
• D. [[3, −1, 2, 5]] — the transformation is recorded as a single row

Answer: (A)

The columns of the matrix are the images of the standard basis vectors: the first column is T(e₁) = (3, −1) and the second column is T(e₂) = (2, 5), giving [[3, 2], [−1, 5]]. Option B is the most common error — placing T(e₁) as a row rather than a column. The column convention follows directly from how matrix-vector multiplication works: Ax = x₁(first column) + x₂(second column), which equals x₁T(e₁) + x₂T(e₂) = T(x₁e₁ + x₂e₂) = T(x).

Question 2

A student claims: 'To determine the matrix of a linear transformation T: Rⁿ → Rᵐ, I need to know how T acts on every possible input vector.' What is the flaw in this reasoning?

• A. Linearity guarantees that T's behavior on any basis determines T's behavior on every vector in the space
• B. The student is correct — you must test infinitely many inputs to fully characterize a transformation
• C. You only need to know T on two vectors regardless of n, since all spaces are at most 2-dimensional in practice
• D. The matrix only needs to encode T's behavior on the zero vector, since all other outputs follow from linearity

Answer: (A)

This is the key insight of the topic. Every vector x can be written as a linear combination of basis vectors: x = x₁b₁ + ... + xₙbₙ. Linearity then gives T(x) = x₁T(b₁) + ... + xₙT(bₙ). So knowing T on n basis vectors is exactly enough information to determine T on all of Rⁿ — the matrix records precisely this information column by column. You don't need to test infinite inputs; the finite basis is sufficient, and this is what makes the matrix representation so powerful.

Question 3

Changing the basis used to represent a linear transformation T changes the matrix that represents T, even though the underlying transformation itself is unchanged.

• T. True
• F. False

Answer: True

The same linear transformation can be represented by different matrices depending on the choice of bases for the domain and codomain. The matrix [T]_B^C encodes the transformation relative to a specific pair of bases B and C. Changing B or C changes the coordinates used, which changes the entries of the matrix — but T itself (as a function mapping vectors to vectors) remains the same geometric object. This is why 'change of basis' is such a fundamental operation: it lets you choose a basis that makes the matrix as simple as possible (e.g., diagonal for an eigendecomposition).

Question 4

If you know a linear transformation T sends two specific vectors u and v to their images T(u) and T(v), you can generally reconstruct the full matrix of T.

• T. True
• F. False

Answer: False

You can reconstruct T from two vectors only if those two vectors form a basis for the domain. If u and v are linearly dependent (one is a scalar multiple of the other), they span only a line, and T's behavior on the rest of the space is completely unconstrained by what you know. For example, knowing T(e₁) tells you nothing about T(e₂). The key condition is that the vectors you evaluate T on must span the domain — which means they must form a basis. Linearity then guarantees the rest.

Question 5

Explain why knowing a linear transformation's effect on a basis is sufficient to determine its effect on every vector in the space.

Model answer: Every vector in Rⁿ can be written uniquely as a linear combination of any basis vectors: x = c₁b₁ + c₂b₂ + ... + cₙbₙ. Because T is linear, it preserves this combination: T(x) = c₁T(b₁) + c₂T(b₂) + ... + cₙT(bₙ). So once we know T(b₁), T(b₂), ..., T(bₙ), we can compute T(x) for any x by expressing x in terms of the basis and applying linearity. The matrix simply stores T(b₁), ..., T(bₙ) as its columns, making this computation automatic via matrix-vector multiplication.

This argument combines two prerequisites — the uniqueness of coordinate representations relative to a basis, and the linearity of T — to show why a finite amount of information (n basis images) determines the transformation on an infinite-dimensional input space. It is the conceptual core of why matrices and linear transformations are literally the same object.