Science:Math Exam Resources/Courses/MATH152/April 2017/Question A 30/Solution 1

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The key idea of this problem is to realize that the plane contains linearly independent vectors; For example, . Therefore, if is a linear transformation that maps into , in fact sends all of to . In particular, we have

for some constants , , and .

On the other hand, we also need to ensure that at least one point of is not mapped to the zero vector: otherwise the image of under would simply be the zero vector instead of all of . In other words, at least one of , , and is non-zero.

Therefore, the matrix of must be of the form

where at least one of , , and is non-zero.