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AlgebraQ&A LibraryPLEASE, I NEED A DETAILED STEP-STEP PROCESS WITH EX-PLANATIONS TO THIS QUESTION.(I have no prior knowledge in Algebra, so be simple as much as possiblefor me.) THANK YOULet K be a field. Prove or disprove each of the following(1) For all ne N, the set of n x n symmetric matrices with entries from K isa subspace of Knxn(2) For all ne N, the set of n x n invertible matrices with entries from K is asubspace of Knxn(3) if T V W is an isomorphism, then T-1 : Walso prove that T1 is 1-1 and ontoV is an isomorphismQuestion

Asked Aug 29, 2019

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I have no strong background in Algebra, so be simple and basic as much as possible for me to grasp each step that you will be provided. I will also rate your answer very high. Thank you so much.

Step 1

Note that, a subspace of a vector space *V* is a subset *H* of *V* that has three properties:

(i). The zero vectors of *V* is in *H*.

(ii). For each *u* and *v* are in *H*, *u* + *v* is in *H.*

(iii). For each *u* in *H* and each scalar *c*, *cu* is in *H*.

Step 2

(ii).

Let *A* and *B* belong to *H*.

Here, *A* and *B* are symmetric. That is,

Step 3

Check closed under ve...

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