Linear Algebra Worksheet 5: Image, Preimage, and Linear Transformation

Finding Images and Preimages of Linear Transformations

• The image of v = (-4, 5, 1) under the linear transformation T(v1, v2, v3) = (4v2 - v1, 2v2 - 3v1, v1 - v3) is T(v) = (-23, -7, -9)
• The preimage of w = (4, 1, -1) under the same transformation is the vector v that satisfies T(v) = w

Determining Linearity

• The function T(x, y) = (x, 1) is not a linear transformation from R2 to R2 because it does not satisfy the linearity property
• A linear transformation must satisfy T(av) = aT(v) and T(u + v) = T(u) + T(v) for all vectors u, v and scalar a

Finding Linear Transformations

• The linear transformation T(x, y) is defined by T(1, 0) = (0, 1) and T(0, 1) = (1, 0), which implies T(x, y) = (y, x) for all (x, y) in R2
• The linear transformation T : R3 → R3 is defined by T(1, 1, 1) = (2, 0, -1), T(0, -1, 2) = (-3, -2, 1), and T(1, 0, 1) = (1, 1, 0)

Finding Outputs of Linear Transformations

• T(2, -1, 1) = (-1, -3, -1) under the linear transformation T : R3 → R3
• T(0, 2, 1) = (5, -2, 0) under the same transformation
• T(2 - 6x + x2) = x2 - x + 1 under the linear transformation T from P2 into P2

Finding Kernels of Linear Transformations

• The kernel of the linear transformation T : P3 → R, where T(a0 + a1 x + a2 x2 + a3 x3) = a0, is the set of all polynomials with constant term equal to 0