Problems:
Linear Algebra : Vector Spaces and Inner Products
1) Let {α1, α2,α 2...........αn} be a basis of an n dimensional vector space over R and A be nxs Matrix .
Let ( β1, β2, β3............... βs) = ( α1, α2, α2...........αn) A
Prove that dim (span { β1, β2, β3............... βs}) = 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
2) Prove that for any λ∈(R^n)' then exists a unique vector α∈R^n such that = {xλ} = xλ , for all ( ε1, ε2, ......... εn)∈ R^n where x.λ is the inner product of R^n.