Problems:
Solving Fundamental Mathematics-Based Questions
1) Determine whether the following sets are basis of ℜ3 or not.
i) {(2,1,-2),(-2,-1,2),(2,4,-4)
ii) {(1,1,1),(1,5,6),(6,2,1)}
2) Let V be the following subset of ℜ4
V = {(x1,x2,x3,x4)∈ℜ4|x1 - x2 - x3 - x4 + 0}
Prove that V is a subspace of ℜ4. Find a abasis of V.
3) Let V1 and V2 be vector spaces. We let V1⊕ V2 be the vector space, Which as a set is the set of all pairs (v1,v2), wher v1∈v2. The addition and multiplication on a scalar is defrined ny.
(v1,v2) + (v'1,v'2) = (v1 + v'1,v2 + v'2),
λ(v1,v2) = (λv1,λv2)
Prove that dim(V1⊕V2) = dim(V1) + dim(V2)
Hint. You can take bases of V1 and V2 and try to construct a basis of V1⊕V2. Alternatively, you can apply the rank-nullity theorem to the linear map p : V1⊕V2 → V2, given by p(v1, v2) = v2.
4) Let U and W be subspaces of vector space V. Prove that
dim(U+W) = dim(U) + dim(W)-dim(U∩W).
Hint. Consider the linear map f : U⊕W → V , given by f(u,w) = u-w, where U⊕W is the same as in the previous problem. Now use the result of the previous problem and the rank-nullity theorem applied on f.
5) Let ƒ : R3→R3 be the linear map given by
ƒ(x1,x2,x3) = (x1 + x2 + x3,x1 - x2,x2 + x3)
Find the matrix corresponding to ƒ in the class { (1,1,0),(-1,1,0),(1,1,1) }.