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Difference between mapping R3 to R2 and the reverse

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Please help. I always have a hard time with Linear Algebra. What's the difference between mapping from R3 into R2 and mapping from R2 into R3?

Why is the following not a linear transformation from R3 into R2?
L(x) = (1 + x1, x2)^T

And why is this one not a linear transformation from R2 into R3?
L(x) = (x1, x2, 1)^T


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Solution Summary

This shows why given mappings are or are not linear transformations.

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The difference between R3->R2 and R2->R3 is that they have different domain and range.
For the mapping ...

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