MATH152 April 2011
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Question B 04 (a)
Let a=[1,2,-1]. Consider the transformation defined by for all x in , where is the cross-product in .
Show that T is a linear transformation.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Recall that a transformation is linear if
for some vectors x, y, and constant c.
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Following the hint, we seek to show the properties of a linear transformation,
For the first property,
where we have used that the cross product is distributive over addition. The second property holds by the compatibility of scalar multiplication with the cross product,
Therefore we have concluded that T is a linear transformation.