Module 5 Quiz: Linear Transformations
Quiz
10 questions on linear transformations, standard matrices, kernel, range, and the Rank-Nullity Theorem.
1
What two properties must a function T satisfy to be a linear transformation?
T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all vectors u, v and scalars c.
2
T(x,y) = (3x - y, x + 2y). What is the standard matrix of T?
A = [3 -1; 1 2]. Columns are T(e1) = (3,1) and T(e2) = (-1,2).
3
What does the kernel of a linear transformation represent geometrically?
The kernel is the set of all vectors that T maps to the zero vector. It represents the "directions that get collapsed" by the transformation.
4
T is one-to-one if and only if ker(T) = ?
{0}. T is one-to-one if and only if the kernel contains only the zero vector (nullity = 0).
5
State the Rank-Nullity Theorem.
For T : R^n → R^m, rank(T) + nullity(T) = n, where n is the dimension of the domain.
6
T : R^5 → R^3 has nullity 3. What is the rank? Is T onto?
rank = 5 - 3 = 2. T is not onto because rank = 2 < 3 = dim(R^3).
7
What is the standard matrix for reflection over the x-axis in R^2?
A = [1 0; 0 -1]. It sends (x,y) to (x, -y).
8
Can a linear transformation T : R^3 → R^2 be one-to-one? Why or why not?
No. By Rank-Nullity, rank + nullity = 3. But rank is at most 2 (the codomain dimension), so nullity is at least 1. A nontrivial kernel means T is not one-to-one.
9
Find the kernel of the transformation with matrix A = [1 -2; -3 6].
Row reduce: [1 -2; 0 0]. Free variable x2 = t, x1 = 2t. Kernel = span{(2, 1)}. Nullity = 1.
10
If T : R^n → R^n is both one-to-one and onto, what can you say about its standard matrix?
The standard matrix A is invertible. Equivalently, det(A) is nonzero, all columns are pivot columns, and rank = n.