Let V and W be finite-dimensional vector spaces and T: V rightarrow W be an isomorphism. Let V0 be a subspace of V. a) Prove that T(V0) is a subspace of W. b) Prove that dim(V0) = dim(T(V0))...
Let θ : R1 → R3 be an isomorphism. We define multiplication between R1 and R2 and between R1 and R3 by the rules xyz = f (y, z)xθ + f (x, y)zθ xθ y = xyθ = f (x, y) (x, y, z ∈ R1) (x, y ∈ R1). Multiplication R2 × R2 → R4 is then determined by...
Let V and W be finite-dimensional vector spaces and T: V rightarrow W be an isomorphism. Let V0 be a subspace of V. a) Prove that T(V0) is a subspace of W. b) Prove that dim(V0) = dim(T(V0)). How to ...