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-221 + Consider the linear transformation T(x1, x2) = (2x1 - x2,-4x1 + 2x2, What is the standard matrix for T? Is the transformation onto? Explain. Is the transformation one-to-one? Explain. If the transformation is not onto, give an example of a vector that is in the range and a vector that is not in the range. If the transformation is not one-to-one, give two preimages for the zero vector.

-221 + Consider the linear transformation T(x1, x2) = (2x1 - x2,-4x1 + 2x2, What is the standard matrix for T? Is the transformation onto? Explain. Is the transformation one-to-one? Explain. If the transformation is not onto, give an example of a vector that is in the range and a vector that is not in the range. If the transformation is not one-to-one, give two preimages for the zero vector.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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+22). Consider the linear transformation T(1, 2) = (2x1x2,-4x1 + 2x2, What is the standard matrix for T? Is the transformation onto? Explain. Is the transformation one-to-one? Explain. If the transformation is not onto, give an example of a vector that is in the range and a vector that is not in the range. If the transformation is not one-to-one, give two preimages for the zero vector.
Transcribed Image Text:+22). Consider the linear transformation T(1, 2) = (2x1x2,-4x1 + 2x2, What is the standard matrix for T? Is the transformation onto? Explain. Is the transformation one-to-one? Explain. If the transformation is not onto, give an example of a vector that is in the range and a vector that is not in the range. If the transformation is not one-to-one, give two preimages for the zero vector.
Expert Solution
Step 1

The given linear transformation T:ℝ2→ℝ3, T(x1,x2)=(2x1-x2,-4x1+2x2,-2x1+x2).

(a) To Find: The standard matrix for T.

(b) To Check: T is onto or not.

(c) To Check: T is one-one or not.

(d) We have to give an example of a vector that is in the range and one that is not in the range.

(e) If the transformation is not one-one then we have to give two examples of preimages of zero vector.

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