PLEASE, 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 isomorphism

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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.

PLEASE, 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 possible
for me.) THANK YOU
Let 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 is
a subspace of Knxn
(2) For all ne N, the set of n x n invertible matrices with entries from K is a
subspace of Knxn
(3) if T V W is an isomorphism, then T-1 : W
also prove that T1 is 1-1 and onto
V is an isomorphism
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PLEASE, 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 possible for me.) THANK YOU Let 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 is a subspace of Knxn (2) For all ne N, the set of n x n invertible matrices with entries from K is a subspace of Knxn (3) if T V W is an isomorphism, then T-1 : W also prove that T1 is 1-1 and onto V is an isomorphism

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Expert Answer

Step 1
  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.

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let K be a filed The set of n x n symmetric matrices. That is, H = n K) To prove: H is subspace of K (). since zero matrix is symmetric Thus, zero matrix belongs to H

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Step 2

(ii).

 Let  A and B belong to H.

Here, A and B are symmetric. That is,

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= A and BF = B

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Step 3

Check closed under ve...

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(4+B AB = A+ B Thus, A B belong to H

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