Invertibility and Isomorphisms
7 important questions on Invertibility and Isomorphisms
What is an invertible transformation?
If T:V->W is linear, and there exists U:W->V linear then U is the inverse of T and can be denoted by T^(-1)
-TU=Iw and UT=Iv
-The inverse of every T is unique
What are some properties of invertible functions T and U?
- inverse of T^(-1) = T
-a function T is invertible iff it is bijective
-if T:V->W is linear, and V and W are finite dimensional vector spaces with the same dimension then, T is invertible iff rank(T)=dim(V)
-if T:V->W is invertible & linear then T(-1):W->V is linear
-if T is invertible & linear from V to W then V is finite-dimensional iff W is finite dimentsional and dim(V)=dim(W)
What are some properties of an invertible square matrix?
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What is an invertible matrix representation?
Then T is invertible iff [T] beta gamma is invertible
- the matrix representation of the inverse of T with respect to beta and gamma = the inverse of the matrix representation of T with respect to beta and gamma
When is a square matrix invertible?
-the inverse of the left transformation of A = the left transformation of the inverse of A
What is an isomorphic linear transformation?
-In that case T is called an isomorphism from V to W
- If V, W are finite-din v.s. , then T is isomorphic from V to W iff dim(V)=dim(W)
When is a vector space F an isomorphism?
The question on the page originate from the summary of the following study material:
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- Never study anything twice again
- Get the grades you hope for
- 100% sure, 100% understanding
