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Questions and Answers
Find the image of v and the preimage of w for the function T(v1, v2, v3) = (4v2 − v1, 2v2 − 3v1, v1 − v3), v = (−4, 5, 1), w = (4, 1, −1)
(b) The preimage of w is (-4, 5, 1) and (a) the image of v is (1, 3, -2)
Determine whether the function T(x, y) = (x, 1) is a linear transformation from R2 to R2
Yes, it is a linear transformation.
Find T(x, y) for (x, y) in R2, given T(1, 0) = (0, 1) and T(0, 1) = (1, 0)
T(x, y) = (y, x)
Find T(2, −1, 1) and T(0, 2, 1) for the linear transformation T : R3 → R3, where T(1, 1, 1) = (2, 0, −1), T(0, −1, 2) = (−3, −2, 1), and T(1, 0, 1) = (1, 1, 0)
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Find T(2 − 6x + x2) for the linear transformation T from P2 into P2, where T(1) = x, T(x) = 1 + x, and T(x2) = 1 + x + x2
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Find the kernel of the linear transformation T : P3 → R, where T(a0 + a1 x + a2 x2 + a3 x3) = a0
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Study Notes
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
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Description
Test your understanding of linear algebra concepts such as image, preimage, linear transformation, kernel, range, rank, and nullity with these exercises. Practice using functions to find the image of a vector and the preimage of a given output vector.
