Below is a five-item multiple-select quiz on Linear Transformations suitable for a weekly assessment. Each item specifies “select all that apply,” is aligned to explicit learning outcomes, and includes an answer key with concise justifications grounded in standard linear algebra results. Citations to authoritative sources are provided.
Question 1. Determining linearity from a rule (Select all that apply)
Let T be a mapping on the indicated domain and codomain. Determine which mappings are linear.
A. T: R^2 → R^2, T(x, y) = (x + 2y, 3x − y)
B. T: R^2 → R^2, T(x, y) = (xy, x + y)
C. T: R^2 → R^3, T(x, y) = (x, y, 1)
D. T: R^2 → R^2, T(x, y) = (0, 0)
E. T: R^2 → R, T(x, y) = |x|
Correct answers: A, D
Rationale: A and D satisfy additivity and homogeneity; B violates linearity due to the product term xy; C violates linearity due to the constant component; E violates homogeneity due to the absolute value (Axler, 2015, Ch. 3; Lay, Lay, & McDonald, 2015, Ch. 1–2).
Learning outcome(s): Identify linear transformations from operational definitions; diagnose violations of linearity.
Question 2. Matrix representation via action on a basis (Select all that apply)
Let T: R^3 → R^2 be linear with T(e1) = (1, 2), T(e2) = (0, −1), and T(e3) = (3, 1), where {e1, e2, e3} is the standard basis of R^3. Determine which statements are necessarily true.
A. The matrix of T in standard coordinates is A = [[1, 0, 3], [2, −1, 1]] (columns are T(e1), T(e2), T(e3)).
B. rank(T) = 2
C. nullity(T) = 1
D. T is onto R^2
E. T is one-to-one
Correct answers: A, B, C, D
Rationale: The matrix columns are T(ei), so A holds. The vectors (1, 2) and (0, −1) are linearly independent, hence rank 2; by rank–nullity, nullity = 3 − 2 = 1; rank equals the dimension of the image, which equals 2 = dim R^2, so T is onto. Nonzero nullity implies T is not injective (Lay et al., 2015, Ch. 2; Strang, 2016, Ch. 3).
Learning outcome(s): Translate action on a basis into a matrix; apply rank–nullity to infer injectivity/surjectivity.
Question 3. Image and kernel from a concrete matrix (Select all that apply)
Let T: R^4 → R^3 be linear with matrix
A = [ [1, 0, 2, 1],
[0, 1, −1, 1],
[0, 0, 0, 0] ].
Determine which statements are true.
A. rank(T) = 2
B. nullity(T) = 1
C. im(T) is a 2-dimensional subspace of R^3
D. T is surjective onto R^3
E. The homogeneous system Ax = 0 has infinitely many solutions
Correct answers: A, C, E
Rationale: Row rank is 2, so rank 2; rank–nullity yields nullity = 4 − 2 = 2 (so B is false); the image is 2-dimensional in R^3 (C true) and thus not all of R^3 (D false); nullity > 0 implies infinitely many solutions to Ax = 0 (E true) (Axler, 2015, Ch. 3; Lay et al., 2015, Ch. 4).
Learning outcome(s): Compute rank and nullity; infer properties of solution sets and surjectivity.
Question 4. Idempotent linear transformations (Select all that apply)
Let T: R^n → R^n be linear and satisfy T^2 = T. Determine which statements must hold.
A. The eigenvalues of T are all in {0, 1}.
B. T is diagonalizable over R.
C. rank(T) = trace(T).
D. T is invertible if and only if T is the identity.
E. ker(T) = im(T).
Correct answers: A, B, C, D
Rationale: The minimal polynomial divides x(x − 1), so eigenvalues are 0 or 1 (A) and the polynomial has no repeated roots, implying diagonalizability over R (B). The trace equals the sum of eigenvalues, which equals the number of 1-eigenvalues; this equals rank(T) (C). If T is invertible and T^2 = T, then multiplying by T^{-1} yields T = I, and conversely I is invertible (D). In general ker(T) and im(T) are complementary subspaces with R^n = ker(T) ⊕ im(T); they need not coincide (E false) (Hoffman & Kunze, 1971, Ch. 6; Axler, 2015, Ch. 5).
Learning outcome(s): Use polynomial constraints to infer spectral properties; connect spectral data to rank/trace; reason about projections and direct sum decompositions.
Question 5. Change of basis and similarity (Select all that apply)
Let T: R^2 → R^2 be linear with standard matrix A. Let B = {v1, v2} be any basis of R^2, and let P be the change-of-basis matrix whose columns are v1 and v2 (expressed in the standard basis). Determine which statements are necessarily true.
A. The matrix of T in basis B is [T]_B = P^{-1} A P.
B. A and [T]_B have the same trace and determinant.
C. If A is diagonalizable, then there exists a basis B such that [T]_B is diagonal.
D. A and [T]_B always have identical columns.
E. If A and [T]_B are similar, then A = [T]_B.
Correct answers: A, B, C
Rationale: Change of basis yields similarity via [T]_B = P^{-1} A P (A). Similar matrices have invariant trace, determinant, and eigenvalues (B). Diagonalizability is equivalent to the existence of a basis consisting of eigenvectors, giving a diagonal representation (C). Columns depend on the basis and generally differ (D false). Similarity does not imply equality (E false) (Strang, 2016, Ch. 5; Axler, 2015, Ch. 5).
Scoring recommendation
- For each item, award full credit only if the learner selects all and only the correct options.
- Alternatively, for partial credit: +1 for each correct option selected, −1 for each incorrect option selected, floor at 0 per item.
References
- Axler, S. (2015). Linear Algebra Done Right (3rd ed.). Springer.
- Hoffman, K., & Kunze, R. (1971). Linear Algebra (2nd ed.). Prentice Hall.
- Lay, D. C., Lay, S. R., & McDonald, J. J. (2015). Linear Algebra and Its Applications (5th ed.). Pearson.
- Strang, G. (2016). Introduction to Linear Algebra (5th ed.). Wellesley-Cambridge Press.
