Please see attached Linear Algebra problem.The proof of Theorem 3.7 used matrix representations and transposition to show that rank(UT)≤rank(T)rank(UT)≤rank(T).How to devise an alternative proof for this inequality, using ordered bases and the Rank-Nullity Theorem (but neither matrix representation nor transposition)? ### Theorem 3.7**Statement:**Let $ T: V \to W $ and $ U: W \to Z $ be linear transformations on finite-dimensional vector spaces $ V, W, $ and $ Z $, and let $ A $ and $ B $ be matrices such that the product $ AB $ is defined. Then(a) $\text{rank}(UT) \leq \text{rank}(U)$.**Proof:**(a) Clearly, $ R(T) \subseteq W $. Hence\[R(UT) = UT(V) = U(T(V)) = U(R(T)) \subseteq U(W) = R(U).\]Thus\[\text{rank}(UT) = \text{dim}(R(UT)) \leq \text{dim}(R(U)) = \text{rank}(U).\]

Name: Please see attached Linear Algebra problem.The proof of Theorem 3.7 us
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Description: Please see attached Linear Algebra problem.The proof of Theorem 3.7 used matrix representations and transposition to show that rank(UT)≤rank(T)rank(UT)≤rank(T).How to devise an alternative proof for this inequality, using ordered bases and the Rank-N

Answered: Theorem 3.7. Let T: V → W and U: W – Z…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Please see attached Linear Algebra problem.

The proof of Theorem 3.7 used matrix representations and transposition to show that rank(UT)≤rank(T)rank(UT)≤rank(T).

How to devise an alternative proof for this inequality, using ordered bases and the Rank-Nullity Theorem (but neither matrix representation nor transposition)?

### Theorem 3.7

**Statement:**
Let $ T: V \to W $ and $ U: W \to Z $ be linear transformations on finite-dimensional vector spaces $ V, W, $ and $ Z $, and let $ A $ and $ B $ be matrices such that the product $ AB $ is defined. Then
(a) $\text{rank}(UT) \leq \text{rank}(U)$.

**Proof:**

(a) Clearly, $ R(T) \subseteq W $. Hence

\[
R(UT) = UT(V) = U(T(V)) = U(R(T)) \subseteq U(W) = R(U).
\]

Thus

\[
\text{rank}(UT) = \text{dim}(R(UT)) \leq \text{dim}(R(U)) = \text{rank}(U).
\]

Branch of mathematics concerned with mathematical structures that are closed under operations like addition and scalar multiplication. It is the study of linear combinations, vector spaces, lines and planes, and some mappings that are used to perform linear transformations. Linear algebra also includes vectors, matrices, and linear functions. It has many applications from mathematical physics to modern algebra and coding theory.

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