linear algebra question

[lotus776]

A recent extra credit question in my linear algebra class asked me to show the linear transformation defined as:

T: M (sub 2,2) -->M (sub 4,1) and also to show the isomorphism.

I know that to show linear transformation I need to display the preservation of addition and scalar multiplication but I don't know how...Should I place both matrices in general form and simply add and then simply multiply by a scalar? Also, isomorphisms are defined as being 1-1 and ONTO but I don't fully understand the question, or how to solve it. Any advice would be greatly apprecited! thank you

-Brent

[fluorescenthippo]

wow your lin ale class is way harder than mine was. i have no idea what you are talking about.

[AlphaWice]

Dude, okay, we have a transformation taking all 2 by 2 matrices to 4 by 1 matrices. To show that addition and multiplication operations are preserved, compare an arbitrary addition (or multiplication) pre transformation and post transformation.

To show the isomorphism, show a bijection between 2x2 and 4x1 matrices (easy.)

[lotus776]

thank you, that is very helpful

[Meromorphic]

[ QUOTE ]
wow your lin ale class is way harder than mine was. i have no idea what you are talking about.

[/ QUOTE ]

It's easy. A vector space is just an R-module, where R is a field, and linear transformation just means R-module homomorphism. Isomorphism of course means what it always does. Did that help clear it up? Would it help to call a linear transformation a morphism in the category of vector spaces?

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