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Solved: Exercises 31 and 32 concern finite-dimensional
Chapter 4, Problem 32E(choose chapter or problem)
Problem 32E
Exercises 31 and 32 concern finite-dimensional vector spaces V and W and a linear transformation
Let H be a nonzero subspace of V , and suppose T is a one-to-one (linear) mapping of V into W . Prove that dim T (H) = D dim H. If T happens to be a one-to-one mapping of V onto W , then dim V = dim W. Isomorphic finite-dimensional vector spaces have the same dimension.
Questions & Answers
QUESTION:
Problem 32E
Exercises 31 and 32 concern finite-dimensional vector spaces V and W and a linear transformation
Let H be a nonzero subspace of V , and suppose T is a one-to-one (linear) mapping of V into W . Prove that dim T (H) = D dim H. If T happens to be a one-to-one mapping of V onto W , then dim V = dim W. Isomorphic finite-dimensional vector spaces have the same dimension.
ANSWER:
Solution 32E
Step 1 of 4
Let be a vector space.
And be a non-zero subspace of a vector space and let be a one-to-one linear mapping.
We have to show that
Since is finite dimensional vector space and is a subspace of
Then is also finite dimensional that has a basis.
