Question Part Points Submissions Used 1 1 /1 2/10 Total 1 /1 Determine whether t

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Question Part Points Submissions Used 1 1/1 2/10 Total 1/1 Determine whether the subset of Mn,n is a subspace of Mn,n with the standard operations of matrix addition and scalar multiplication. The set of all n × n matrices with integer entries

Ok, I am having a very, very difficult time understanding vector spaces and subspaces. Can someone please, try to assist me in understanding the method(s) I need to go about doing to help understand how vector spaces/subspaces work?? Ive had no lucky anywhere else, or with any other tutor, unfortunately. I will for sure give full points if its clear.. thank you!
Question Part Points Submissions Used 1 1/1 2/10 Total 1/1 Question Part Points Submissions Used 1 1/1 2/10 Total 1/1 Mn,n Mn,n The set of all n × n matrices with integer entries

Ok, I am having a very, very difficult time understanding vector spaces and subspaces. Can someone please, try to assist me in understanding the method(s) I need to go about doing to help understand how vector spaces/subspaces work?? Ive had no lucky anywhere else, or with any other tutor, unfortunately. I will for sure give full points if its clear.. thank you!
Question Part Points Submissions Used

Explanation / Answer

For vector space satisfy these axioms :

(1) Commutative law: For all vectors u and v in V, u + v = v + u

(2) Associative law: For all vectors u, v, w in V, u + (v + w) = (u + v) + w

(3) Additive identity: The set V contains an additive identity element, denoted by 0, such that for any vector v in V, 0 + v = v and v + 0 = v.

(4) Additive inverses: For each vector v in V, the equations v + x = 0 and x + v = 0 have a solution x in V, called an additive inverse of v, and denoted by - v.

(5) Distributive law: For all real numbers c and all vectors u, v in V, c · (u + v) = c · u + c · v

(6) Distributive law: For all real numbers c, d and all vectors v in V, (c+d) · v = c · v + d · v

(7) Associative law: For all real numbers c,d and all vectors v in V, c · (d · v) = (cd) · v

(8) Unitary law: For all vectors v in V, 1 · v = v

For subspace check for only these 3 conditions:

1. W is nonempty: The zero vector belongs to W.

2.Closure under +: If u and v are any vectors in W, then u + v is in W.

3.Closure under ·: If v is any vector in W, and c is any real number, then c · v is in W.

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