Linear Algebra Quiz: Linear Transformation Definition

What this quiz covers

This quiz focuses on Linear Transformation Definition, giving you a quick way to practice the rules, question types, and explanations that matter most for Linear Algebra.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

Question 1

1. $T(p(x)) = p(1) + 1$
2. $T(p(x)) = \int_0^1 (p(x))^2 dx$
3. $T(p(x)) = p(0) \cdot p'(0)$
4. $T(p(x)) = p(2) - p'(1)$ (correct answer)

Question 2

1. The condition is sufficient to prove linearity, as it follows from the homogeneity property $T(c\vec{u}) = cT(\vec{u})$ by setting $c=0$.
2. The condition is both necessary and sufficient, meaning any transformation mapping the zero vector to the zero vector is linear.
3. The condition is necessary for linearity, but it is not sufficient, as non-linear transformations exist that also map the zero vector to the zero vector. (correct answer)
4. The condition is not necessary for linearity, as some linear transformations, such as translations, do not map the zero vector to the zero vector.

Question 3

1. For $c = -2$ and $\vec{u} = (1, 3)$, we have $T(c\vec{u}) = T(-2, -6) = (-2, 6)$, while $cT(\vec{u}) = -2T(1, 3) = -2(1, 3) = (-2, -6)$. (correct answer)
2. For $\vec{u} = (1, 2)$ and $\vec{v} = (3, 4)$, $T(\vec{u}+\vec{v}) = T(4, 6) = (4, 6)$ and $T(\vec{u}) + T(\vec{v}) = (1, 2) + (3, 4) = (4, 6)$, verifying additivity.
3. $T(1, 2) = (1, 2)$ and $T(1, -2) = (1, 2)$, which shows that $T$ is not an injective (one-to-one) function.
4. $T(0, 0) = (0, |0|) = (0, 0)$, which satisfies a necessary condition for linearity, but does not prove it.

Question 4

1. $T: M_{2 \times 2} \to \mathbb{R}$ defined by $T(A) = \det(A)$
2. $T: M_{2 \times 2} \to \mathbb{R}$ defined by $T(A) = (A^2)_{11}$, the entry in the first row and first column of $A^2$
3. $T: M_{2 \times 2} \to M_{2 \times 2}$ defined by $T(A) = A + I$, where $I$ is the identity matrix
4. $T: M_{2 \times 2} \to \mathbb{R}$ defined by $T(A) = \mathrm{trace}(A)$ (correct answer)

Question 5

1. $F(\vec{v}) = (T(\vec{v}))^2$ (correct answer)
2. $F(\vec{v}) = T(\vec{v}) + T(-\vec{v})$
3. $F(\vec{v}) = 2T(\vec{v})$
4. $F(\vec{v}) = T(\vec{v} - \vec{v}_0) + T(\vec{v}_0)$ for a fixed vector $\vec{v}_0 \in \mathbb{R}^2$

Question 6

1. $\begin{bmatrix} 6 \\ 5 \\ 11 \end{bmatrix}$
2. $\begin{bmatrix} 2 \\ 3 \\ 4 \end{bmatrix}$
3. $\begin{bmatrix} 6 \\ -11 \\ 7 \end{bmatrix}$ (correct answer)
4. $\begin{bmatrix} 6 \\ -8 \\ 2 \end{bmatrix}$

Question 7

1. $T(c\vec{u} + d\vec{v}) = cT(\vec{u}) + dT(\vec{v})$ for all vectors $\vec{u}, \vec{v} \in V$ and all scalars $c, d$. (correct answer)
2. $T(\vec{u} - \vec{v}) = T(\vec{u}) - T(\vec{v})$ for all vectors $\vec{u}, \vec{v} \in V$.
3. $T(c\vec{u}) = cT(\vec{u})$ for all vectors $\vec{u} \in V$ and all scalars $c$.
4. $T(\vec{u} + \vec{v}) = T(\vec{u}) + T(\vec{v})$ for all vectors $\vec{u}, \vec{v} \in V$.

Question 8

1. The matrix $A$ must be the zero matrix.
2. The vector $\vec{b}$ must be the zero vector. (correct answer)
3. The matrix $A$ must be invertible.
4. The vector $\vec{b}$ must be in the column space of $A$.

Question 9

1. If $\{\vec{v}_1, \dots, \vec{v}_k\}$ is a linearly independent set in $V$, then $\{T(\vec{v}_1), \dots, T(\vec{v}_k)\}$ is a linearly independent set in $W$.
2. The dimension of the domain $V$ must be less than or equal to the dimension of the codomain $W$.
3. For any vectors $\vec{u}, \vec{v} \in V$, the image of the line segment connecting them is the line segment connecting $T(\vec{u})$ and $T(\vec{v})$. (correct answer)
4. The transformation $T$ must be surjective (onto).

Question 10

1. The student failed to verify the additivity property $G(\mathbf{u} + \mathbf{v}) = G(\mathbf{u}) + G(\mathbf{v})$, which is the more important condition
2. The formula $G(c\mathbf{v}) = cG(\mathbf{v}) + 5(1-c)$ is mathematically incorrect and doesn't hold for this function
3. Linear transformations must map from $\mathbb{R}^n$ to $\mathbb{R}^n$, so functions with codomain $\mathbb{R}$ cannot be linear
4. The presence of the constant term 5 means $G$ is affine, not linear, and the modified homogeneity condition is irrelevant to linearity (correct answer)

Question 11

1. $H$ must map linearly independent sets to linearly independent sets and preserve basis relationships
2. $H$ must be continuous and differentiable with respect to the standard topology on both vector spaces
3. $H$ must preserve the inner product structure: $\langle H(\mathbf{u}), H(\mathbf{v}) \rangle = \langle \mathbf{u}, \mathbf{v} \rangle$ for all $\mathbf{u}, \mathbf{v}$
4. $H$ must also satisfy $H(c\mathbf{v}) = cH(\mathbf{v})$ for all scalars $c$ and vectors $\mathbf{v} \in V$ (correct answer)

Question 12

1. It demonstrates that $T$ satisfies the additivity property but says nothing about scalar multiplication
2. It completely proves that $T$ is linear since the chosen vectors span a 2-dimensional subspace of $\mathbb{R}^3$
3. It confirms linearity for one specific case, but linearity requires this property to hold for all possible vectors and scalars (correct answer)
4. It shows that $T$ is linear on the subspace spanned by $\mathbf{u}$ and $\mathbf{v}$, which is sufficient for global linearity

Question 13

1. Demonstrate that $R$ has a matrix representation and that matrix multiplication always produces linear transformations
2. Verify that $R$ preserves lengths and angles, which are the fundamental requirements for linear transformations
3. Show that $R(c\mathbf{v}) = cR(\mathbf{v})$ and $R(\mathbf{u} + \mathbf{v}) = R(\mathbf{u}) + R(\mathbf{v})$ using geometric properties of rotation (correct answer)
4. Show that $R$ maps the standard basis vectors to orthogonal unit vectors, confirming orthogonal transformation properties

Question 14

1. The student must confirm that $L$ maps the zero vector to the zero vector, which is a necessary condition for linearity
2. The student must verify that $L(a\mathbf{u} + b\mathbf{v}) = aL(\mathbf{u}) + bL(\mathbf{v})$ for arbitrary vectors and scalars, not just basis vectors with specific scalars (correct answer)
3. The student must verify that $L$ preserves linear independence by showing that linearly independent inputs produce linearly independent outputs
4. The student must demonstrate that the transformation is invertible by constructing the inverse function $L^{-1}$

Question 15

1. Every linear transformation can be represented this way, and every such matrix representation defines a linear transformation (correct answer)
2. Only bijective linear transformations can be represented as matrix multiplication, while others require additional constant terms
3. Matrix representation is possible only when the domain and codomain have the same finite dimension
4. The matrix representation exists if and only if the transformation preserves orthogonality and has determinant equal to ±1

Question 16

1. $T(x, y) = (x+y, x-1)$
2. $T(x, y) = (2x - 3y, x + 5y)$ (correct answer)
3. $T(x, y) = (xy, y)$
4. $T(x, y) = (x^2, y^2)$

Question 17

1. The given property must hold for all vectors $\mathbf{u}, \mathbf{v} \in \mathbb{R}^3$ and all scalars, not just the specific case shown (correct answer)
2. $F$ must also satisfy $F(\mathbf{u} \times \mathbf{v}) = F(\mathbf{u}) \times F(\mathbf{v})$ for the cross product operation between vectors
3. The function must be continuous at the origin and differentiable throughout its domain with constant derivative
4. $F$ must preserve orthogonality, meaning $\mathbf{u} \perp \mathbf{v}$ implies $F(\mathbf{u}) \perp F(\mathbf{v})$ for all input vectors

Question 18

1. A rotation of $30^\circ$ counter-clockwise about the origin.
2. A reflection across the line $y=x$.
3. A projection onto the x-axis.
4. A translation by the vector $\begin{bmatrix} 1 \\ 1 \end{bmatrix}$. (correct answer)

Question 19

1. $P(3, -6) = (0, 0)$ because the vector $(3, -6)$ is orthogonal to the line $y = x$ and linear projections map orthogonal vectors to zero
2. $P(3, -6) = (-1.5, -1.5)$ because linearity requires that the projection formula $P(x,y) = \left(\frac{x+y}{2}, \frac{x+y}{2}\right)$ applies consistently (correct answer)
3. $P(3, -6) = (3, 3)$ because linear transformations preserve the $x$-component when projecting onto lines through the origin
4. $P(3, -6)$ cannot be determined without additional information about how $P$ affects the standard basis vectors

Question 20

1. $T$ preserves addition and scalar multiplication independently for all vectors in the domain (correct answer)
2. $T$ maps the zero vector to itself and preserves the magnitude of all input vectors
3. $T$ is bijective and preserves the dot product between any two vectors in the domain
4. $T$ commutes with matrix multiplication and preserves linear independence of basis vectors