Let T: V
V and S: V
V and R: V
V be three linear operators on V. Suppose we have
T
S= S
R , Then prove ker(S) is an invariant subspace for R .
Let V be a finite-dimensional vector space and let T L(V) be an operator. In this problem you show that there is a nonzero polynomial such that p(T) = 0. (a) What is 0 in this context? A polynomial? A linear map? An element of V? (b) Define by . Prove that is a linear map. (c) Prove that if where V is infinite-dimensional and W is finite-dimensional, then S cannot be injective. (d) Use the preceding parts to prove...
Note: In the following, if is a set and both and are positive integers, then matrices with entries from . The problem below has many applications. If is a linear map from complex vector space to itself, and is an eigenvalue of , then is a simple eigenvalue of if . 1. Suppose is a vector space of dimension over field where you may assume that is either or , and let be a linear map from to . Show...
Problem 3. Let V and W be vector spaces, let T : V -> W be a linear transformation, and suppose U is a subspace of W (a) Recall that the inverse image of U under T is the set T-1 U] := {VE V : T(v) E U). Prove that T-[U] is a subspace of V (b) Show that U nim(T) is a subspace of W, and then without using the Rank-Nullity Theorem, prove that dim(T-1[U]) = dim(Unin (T))...
Let n, and let n be a reduced residue. Let r = odd(). Prove that if r = st for positive integers s and t, then old(t) = s. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
for a linear operator T ∈ L(V), V is finite-dimensional. let C={r(T)(v): r(x) ∈ F[x], v non zero} show that C is an invariant of T for the subspace of V.
Let T : C([0, 1]) → R be a (not necessarily bounded) linear functional. Show that T is positive if and only if = (here 1 denotes the constant function [0, 1] → R, x → 1). We were unable to transcribe this imageWe were unable to transcribe this image
Let T: be defined as . Prove or disprove that can be written as the sum of two one-dimensional, T-invariant subspaces. IR IR We were unable to transcribe this imageWe were unable to transcribe this image IR IR
Using FTLM. a) Let . Use linear algebra to prove that there is a polynomial such that p + p' - 3p'' = q. Hint: consider the map defined by Tp: p + p' - 3p'', and use FTLM. b) Let be distinct elements of . Let be any elements of . Use linear algebra to prove that there is a such that Hint: consider the map defined by . You can use any facts from algebra about the solution...
could somone plz help with #4 3. If T : p → W is a linear transformation, then T is one-to-one if and only if ker T = {0} 4. Prove that if T:V-is a linear transformation and W is a subspace of V, then the image of W'is a subspace of V" 3. If T : p → W is a linear transformation, then T is one-to-one if and only if ker T = {0} 4. Prove that if...
Let V be a Hilbert space. Let S1 and S2 be two hyperplanes in V defined by Let be given. We consider the projection of y onto , i.e., the solution of (1) (a) Prove that is a plane, i.e., if , then for any . (b) Prove that z is a solution of (1) if and only if and (2) (c) Find an explicit solution of (1). ( d) Prove the solution found in part (c) is unique. We...