4. (10 pts) Let De Mn (R) be invertible and define the map T: Mn(R) →...
Let and let T : R4 → R4 be the map T(v) = v-2 proju,V-2 proju, w (a) Show that T is a linear transformation. (b) Find T] and (c) Show that T is invertible and find T-1 Let and let T : R4 → R4 be the map T(v) = v-2 proju,V-2 proju, w (a) Show that T is a linear transformation. (b) Find T] and (c) Show that T is invertible and find T-1
Let M be a 8:27 AM right R-module, N be an (R,T)-bimodule, and L be a left T-module. Let e: (MN)* L M R (NB, L) be given by e (moon, e) = m (nol). Let m.con, mone MORN, and lEl. Prove e (lm, BR.) + (m₂ Ore), d)= e(m, on, d) + (mon, e). This is the proof I'm working on. I need to show the map I've defined (and which is defined towards the middle of the proof)...
1. Let T: R2 – R? be the map "reflection in the line y = x"—you may assume this T is linear, let Eº be the standard basis of R2 and let B be the basis given by B = a) On the graph below, draw a line (colored if possible) joining each of the points each of the points (-). (). (1) and () woits image to its image under the map T. y = x b) Find the...
Consider a linear space P2(R) with the standard basis S- {1,t,t, t 3). a. Describe the isomorphism P R sending p(t) ps b. Show that B [t - 1,t + 1,t2 +t, t3) is another basis for P3 (R). c. Let p(t) 32t4t3. Find p. d. Show that the map P R4 sending p(t)-, рв is an isomorphism.
Problem 4. Let n E N, and let V be an n-dimensional vector space. Let(, ,): V × V → R be an nner product on V (a) Prove that there exists an isomorphism T: V -R" such that (b) Is the isomorphism T you found in part (a) unique? Give a proof or a counterexample. (c) Let A be an n × n symmetric matrix such that T A > 0 for all nonzero ERT. Show that there exists...
2. Let R be an integral domain containing a field K as a unital subring. (a) Prove that R is a K-vector space (using addition and multiplication in R). (b) Let a be a nonzero element of R. Show that the map is an injective K-linear transformation and is an isomorphism if and only if is invertible as an element of R. (c) Suppose that R is finite dimensional as a K-vector space. Prove that R is a field.
5. For t ER, define the evaluation map evt : Pn(R) + R given by evt(p(x)) = p(t). Here we consider R as the vector space R1. (a) Prove evt is a linear map. (b) For part (b), let n= 4. Write down a polynomial p e ker(ev3). (c) For any t, the set of polynomials Ut = {p E Pn(R) : p(t) = 0} is a subspace. What is the dimension of Ut (in terms of n)? Justify your...
Justify statement 1-4 and explain why. If a matrix A is invertible, then all the eigenvalues of A are nonzero. If two linear maps have the same characteristic polynomial, then they always have the same Jordan canonical form. If a linear map from the vector space P of all polynomials to itself is injective, then it is an isomorphism. If W, and W2 are subspaces of a vector space V, then the projection T: W W 2 → W, i.e.,...
4. (10 points) Let T:P2 + M2x2(R) be the linear map Tax+bæ + c) =( and 1/1 0 1 B' = 0 1 0 0 O 0) (10) be the standard basis for the codom(T). Using 1 any basis B for P2, compute TB.
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