Linear Transformation Worksheet
Grade 12 Math worksheet on linear transformations, including matrix representation, kernel, range, injectivity, and surjectivity.
Includes
Standards
Topics
Linear Transformation Worksheet
Name:
Date:
Score:
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.
Related Worksheets
Perfect Square Trinomials
This worksheet focuses on identifying, factoring, and constructing perfect square trinomials, essential skills for Grade 12 Algebra.
Equations with Infinite and No Solutions
Explore linear equations that result in infinite solutions (identities) or no solutions (contradictions) at a Grade 12 level.
Grade 12 Algebra: Standard Form
A Grade 12 math worksheet focusing on converting between different forms of linear equations, identifying slopes and intercepts, and graphing linear equations in standard form.
Characteristics of Parabolas Worksheet
Explore and identify the key characteristics of parabolas, including vertex, focus, directrix, and axis of symmetry.
Solving Equations with Variables on Both Sides
This worksheet provides practice on solving linear equations where variables appear on both sides of the equals sign.
Evaluating Algebraic Expressions
This worksheet focuses on evaluating complex algebraic expressions involving various operations and functions, suitable for Grade 12 students.
Graphing Parabolas Worksheet
A Grade 12 math worksheet on graphing parabolas, covering standard form, vertex form, and intercepts.
Grade 12 Sigma Notation Worksheet
A comprehensive worksheet for Grade 12 students covering sigma notation, including evaluating sums, expanding series, and writing expressions in sigma notation.