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Quiz: Linear Transformations

Test your understanding of linear transformations and their properties.

1. A transformation \(T\) is linear if it satisfies:

  1. \(T(\mathbf{x} + \mathbf{y}) = T(\mathbf{x}) + T(\mathbf{y})\) and \(T(c\mathbf{x}) = cT(\mathbf{x})\)
  2. \(T(\mathbf{x}) = \mathbf{x}\) for all \(\mathbf{x}\)
  3. \(T(\mathbf{x}) \cdot T(\mathbf{y}) = T(\mathbf{x} \cdot \mathbf{y})\)
  4. \(T(\mathbf{0}) \neq \mathbf{0}\)
Show Answer

The correct answer is A. A linear transformation satisfies two properties: additivity \(T(\mathbf{x} + \mathbf{y}) = T(\mathbf{x}) + T(\mathbf{y})\) and homogeneity \(T(c\mathbf{x}) = cT(\mathbf{x})\). These can be combined as \(T(a\mathbf{x} + b\mathbf{y}) = aT(\mathbf{x}) + bT(\mathbf{y})\).

Concept Tested: Linear Transformation

2. Every linear transformation can be represented by:

  1. A scalar
  2. A vector
  3. A matrix
  4. A polynomial
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The correct answer is C. Every linear transformation between finite-dimensional vector spaces can be represented by a matrix. For \(T: \mathbb{R}^n \to \mathbb{R}^m\), there exists a unique \(m \times n\) matrix \(A\) such that \(T(\mathbf{x}) = A\mathbf{x}\).

Concept Tested: Matrix Representation

3. What is the kernel (null space) of a linear transformation?

  1. The set of all possible outputs
  2. The set of all inputs that map to the zero vector
  3. The identity transformation
  4. The inverse transformation
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The correct answer is B. The kernel (or null space) of a linear transformation \(T\) is the set of all vectors \(\mathbf{x}\) such that \(T(\mathbf{x}) = \mathbf{0}\). It measures the "collapse" in the transformation.

Concept Tested: Kernel

4. The image (range) of a linear transformation is:

  1. The set of all inputs
  2. The set of all possible outputs
  3. The zero vector
  4. The inverse of the kernel
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The correct answer is B. The image (or range) of a linear transformation is the set of all possible outputs—all vectors that can be reached by applying the transformation to some input. It equals the column space of the matrix representation.

Concept Tested: Image

5. A rotation in 2D by angle \(\theta\) counterclockwise is represented by:

  1. \(\begin{bmatrix} \cos\theta & \sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\)
  2. \(\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\)
  3. \(\begin{bmatrix} \sin\theta & \cos\theta \\ -\cos\theta & \sin\theta \end{bmatrix}\)
  4. \(\begin{bmatrix} \theta & 0 \\ 0 & \theta \end{bmatrix}\)
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The correct answer is B. The standard 2D rotation matrix for counterclockwise rotation by angle \(\theta\) is \(\begin{bmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{bmatrix}\). This preserves lengths and rotates vectors by the specified angle.

Concept Tested: Rotation Transformation

6. A linear transformation is injective (one-to-one) if and only if:

  1. Its image is the entire codomain
  2. Its kernel contains only the zero vector
  3. It is represented by a square matrix
  4. It preserves the dot product
Show Answer

The correct answer is B. A linear transformation is injective if and only if its kernel is trivial (contains only \(\mathbf{0}\)). This means different inputs always produce different outputs—no two distinct vectors map to the same output.

Concept Tested: Injective Transformation

7. Which transformation scales a vector by 2 in the x-direction only?

  1. \(\begin{bmatrix} 2 & 0 \\ 0 & 2 \end{bmatrix}\)
  2. \(\begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\)
  3. \(\begin{bmatrix} 1 & 2 \\ 0 & 1 \end{bmatrix}\)
  4. \(\begin{bmatrix} 0 & 2 \\ 2 & 0 \end{bmatrix}\)
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The correct answer is B. The matrix \(\begin{bmatrix} 2 & 0 \\ 0 & 1 \end{bmatrix}\) scales the x-component by 2 while leaving the y-component unchanged. This is a non-uniform scaling (stretching) transformation.

Concept Tested: Scaling Transformation

8. The composition of two linear transformations \(T_1\) and \(T_2\) corresponds to:

  1. Adding their matrices
  2. Multiplying their matrices
  3. Finding the inverse of their matrices
  4. Taking the transpose of their matrices
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The correct answer is B. The composition of linear transformations corresponds to matrix multiplication. If \(T_1\) is represented by \(A\) and \(T_2\) by \(B\), then \(T_2 \circ T_1\) (apply \(T_1\) first, then \(T_2\)) is represented by \(BA\).

Concept Tested: Composition of Transformations

9. A reflection across the x-axis is represented by:

  1. \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\)
  2. \(\begin{bmatrix} -1 & 0 \\ 0 & 1 \end{bmatrix}\)
  3. \(\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\)
  4. \(\begin{bmatrix} -1 & 0 \\ 0 & -1 \end{bmatrix}\)
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The correct answer is A. The matrix \(\begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\) reflects vectors across the x-axis by negating the y-component while preserving the x-component.

Concept Tested: Reflection Transformation

10. A shear transformation in the x-direction:

  1. Rotates vectors around the origin
  2. Scales all vectors uniformly
  3. Shifts x-coordinates proportionally to y-coordinates
  4. Projects vectors onto the x-axis
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The correct answer is C. A shear in the x-direction shifts x-coordinates by an amount proportional to the y-coordinate, represented by \(\begin{bmatrix} 1 & k \\ 0 & 1 \end{bmatrix}\). This slants rectangles into parallelograms while preserving area.

Concept Tested: Shear Transformation