Problem 4. Suppose L: V → W is a linear map, with L(V) = W. Prove...
4. Let L: V→ W be a linear map. Let w be an element of W. Let uo be an ele- ment of V such that LvO-w. Show that any solution of the equation L(X)-w is of type uo + u, where u is an element of the kernel of L.
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...
Problem 1. Prove that the composition of injective linear maps, when it is defined, yields injective linear map an Problem 2. Prove that if V = span(v1....,) and fe L(V,W) is surjec- Problem 1. Prove that the composition of injective linear maps, when it is defined, yields injective linear map an Problem 2. Prove that if V = span(v1....,) and fe L(V,W) is surjec-
Prove the following → V such that (a) If T:V + W is linear and injective, then there exists a linear map S: W ST = I. (b) If S: W → V is linear and surjective, then there exists a linear map T:V ST = 1. W such that
Let f: V W be a linear map. Prove that f(0) = 0, i.e., any linear transformation maps zero vector to zero vector.
Let T :V → W be an isomorphism. Prove that if {ū1, ū2, ..., ūn} is a linearly independent set in W, then the preimages of {ū1, ū2, ... , ūn} is a linearly independent set in V.
2. Prove that: For a one-to-one linear transformation L: V + W, the dimension of V is the same as the dimension of L(V). . ..... .... . ... .
Question #4 Consider the linear map, Prove that L^n x goes to 0 for all x in R^2. prove that if x does not lie on the y axis then the orbit of x tends to 0 tangentially to the x -axis. 4. Consider the linear map 0 L(x) = X. Prove that L"X → 0 for all x E R2. Prove that, if x does not lie on the y-axis, then the orbit of x tends to 0 tangentially...
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
(1) Suppose that V and W are both finite dimensional vector spaces. Prove that there exists a surjective linear map from V onto W if and only if Dim(W) Dim(V)