Let f: V W be a linear map. Prove that f(0) = 0, i.e., any linear...
1. let V be a vector space and T an operator on V (i.e., a linear map T: V--> V). Suppose that T^2 - 5T +6I = 0, where I is the identity operator and 0 stands for the zero operator ... Read Section 3.E and 3.F V) 1. Let V be a vector space and T an operator on V (i.e., a linear map T: V -» Suppose that T2 - 5T + 61 = 0, where I is...
Problem 5. Let V and W be vector spaces, and suppose that B (vi, ..., Vn) is a basis of V a) Prove that for every function f : B → W, there exists a linear transformation T: V → W such that T(v;)-f(7) for all vEB (b) Prove that for any two linear transformations S : V → W and T : V → W, if S(6) = T(6) for all ï, B, then S = T (c) Prove...
Vectors pure and applied Exercise 11.5.9 Let U and V be finite dimensional spaces over F and let θ : U linear map. v be a (i) Show that o is injective if and only if, given any finite dimensional vector space W map : V W such that over F and given any linear map α : U-+ W, there is a linear (ii) Show that θ is surjective if and only if, given any finite dimensional vector space...
Let V and W be vector spaces over F, and let f: V W be a linear transformation. (a) Prove that f is one-to-one if and only if f carries linearly independent (b) Suppose that f is one-to-one and that S is a subset of V. Prove that subsets of V to linearlv independent subsets of W S is linearly independent if and only if (S) is linearly independent.
Let V and W be a vector spaces over F and T ∈ L(V, W) be invertible. Prove that T-1 is also linear map from W to V . Please show all steps, thank you
Exercise 12.6.3 Let V and W be finite dimensional vector spaces over F, let U be a subspace of V and let α : V-+ W be a surjective linear map, which of the following statements are true and which may be false? Give proofs or counterexamples O W such that β(v)-α(v) if v E U, and β(v) (i) There exists a linear map β : V- otherwise (ii) There exists a linear map γ : W-> V such that...
Two questions,please! 7. Assume C is a linear code. Prove that G is a generator matrix for C if and only if the columns of G form a basis of C 8. Let V. W U be vector spaces over F of finite dimension and φ: V → W, t : W → U linear maps. Prove that Im(φ)-ker( ) holds if and only if ψφ-0 and dimF1m(φ)-dimF kere). 7. Assume C is a linear code. Prove that G is...
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> W be a linear transformation. (a) Prove that for every pair of ordered bases B = exists a unique m x n matrix A such that [T(E)]c = A[r3 for all e V. The matrix A is called the (B,C)-matrix of T, written A = c[T]b. (b) For each n E N, let Pm be the vector space of...
Problem 3. Let V and W be vector spaces of dimensions n and m, respectively, and let T : V -> V be a linear transformation (a) Prove that for every pair of ordered bases B = (Ti,...,T,) of V and C = (Wi, ..., Wm) of W, then exists a unique (B, C)-matrix of T, written A = c[T]g. (b) For each n e N, let Pn be the vector space of polynomials of degree at mostn in the...
Let T be a linear operator on F2. Prove that if v f 0 is not an eigenvector for T, then v is a cyclic vector for T. Conclude that either T has a cyclic vector T is a scalar multiple of the identity.