3. Linear Algebra

Linear Transformations — Quiz

Test your understanding of linear transformations with 5 practice questions.

Practice Questions

Question 1

Let $T: V \to W$ be a linear transformation. If $B = \{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\}$ is a basis for $V$, which of the following statements is true regarding the image of $T$?

Question 2

Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be a linear transformation defined by $T(x, y, z) = (x + y, y + z, x - z)$. Find a basis for the kernel of $T$.

Question 3

Consider the linear transformation $T: \mathbb{R}^2 \to \mathbb{R}^2$ that first reflects a vector across the line $y = x$ and then scales it by a factor of 2. What is the matrix representation of $T$ with respect to the standard basis?

Question 4

Let $T: V \to W$ be a linear transformation. If $\text{dim}(V) = 7$ and $\text{dim}(\text{Im}(T)) = 4$, what is the dimension of the kernel of $T$?

Question 5

Let $T: P_2(\mathbb{R}) \to P_1(\mathbb{R})$ be a linear transformation defined by $T(ax^2 + bx + c) = (a - b)x + (b - c)$, where $P_k(\mathbb{R})$ is the vector space of polynomials of degree at most $k$. Find the matrix representation of $T$ with respect to the standard bases $B_2 = \{x^2, x, 1\}$ for $P_2(\mathbb{R})$ and $B_1 = \{x, 1\}$ for $P_1(\mathbb{R})$.