Linear Transformation Worksheet
Grade 12 Math worksheet on linear transformations, including matrix representation, kernel, range, injectivity, and surjectivity.
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Linear Transformation Worksheet
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Read each question carefully and provide your answers in the space provided. Show all your work for full credit.
1. Let T: ℝ² → ℝ² be a linear transformation defined by T(x, y) = (2x + 3y, x - y). Find the standard matrix representation of T.
2. Which of the following transformations from ℝ² to ℝ² is a linear transformation?
T(x, y) = (x + 1, y)
T(x, y) = (xy, y)
T(x, y) = (2x, -y)
T(x, y) = (|x|, y)
3. The kernel of a linear transformation T: V → W is the set of all vectors v in V such that T(v) = . The range of T is the set of all vectors w in W such that w = for some v in V.
4. A linear transformation T is injective (one-to-one) if and only if its kernel contains only the zero vector.
True
False
5. Let T₁: ℝ² → ℝ² be defined by T₁(x, y) = (x + y, x - y) and T₂: ℝ² → ℝ² be defined by T₂(x, y) = (2x, 3y). Find the matrix representation of the composition (T₂ ∘ T₁).
6. Describe the geometric effect of the linear transformation T(x, y) = (-x, y) on a vector in ℝ².