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Linear Algebra : Vector Spaces and Inner Products

1) Let { 1, 2, 2........... n} be a basis of an n dimensional vector space over R and A be n Matrix .
Let ( 1, 2, 3............... s) = ( 1, 2, 2........... n) A
Prove that dim (span { 1, 2, 3............... s}) = Rank (A).

2) Let V1 be the solution space of x1 +x2 + x3............+xn = 0
let V2 be the solution space of x1 =x2 = x3............=xn.
Prove that R^n (R to the power n) = V1+V2

3) Prove that for any (R^n)' then exists a unique vector R^n such that
= x for all ( 1, 2, ......... n) R^n
where x is the inner product of R^n.

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Proofs involving vector spaces and inner products are provided. The solution is detailed and well presented. The response received a rating of "5" from the student who originally posted the question.

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