Question 1.2 Let T : R3 ? R2 be a linear transformation given by T (x) = Ax, where 1 0 2 -1 1 5 1) Find a basis for the kernel of T. 2) Determine the dimension of the kernel of T 3) Find a basis for the image(range) of T. 4) Determine the dimension of the image(range) of T. 5) Determine if it is a surjection or injection or both. 2 6) Determine whether or not v |0|...
Let LA be the linear map from R2 to R2 defined by LA (i) = Av, and let LB be the linear map from R2 to IR2 defined by LB(T)-Bv where A -6 36 -1 6 and B-(1 0 The composition LA O LB is again a linear map Lc determined by a (2 x 2)-matrix C, such that Calculate C C- Let LA be the linear map from R2 to R2 defined by LA (i) = Av, and let...
LI), Let T be a linear map from R5 to R3 with i. Show that T must be surjective. ii. Show that there can be no such T from R to R2
Let T be a linear map from R3[z] to R2[z] defined as (T p)(z) = p'(z). Find the matrix of T in the basis: 4 points] Let T be a linear map from Rals] to R12] defined as (TP)(z) = p,(z). Find the matrix of T in the basis: in R2[-]; ~ _ s, r2(z) (z-s)2 in R2 [2], where t and 8 are real numbers. T1(2 Find coordinates of Tp in the basis lo, 1, 12 (if p is...
R4, and the set V of vectors i (4 points. Consider a linear transformation T: R3 in R3 such that T(T) = . Is V a subspace of R3? (8 points.) Suppose a matrix A is 6 x 4. Explain each of your answers in one sentence. If, looking at A, you can easily tell it has at least one row which is a linear com- bination of some of the other rows, what does that tell you about the...
SF78. Consider the linear map T : Rn → Rm defined by T(v) = Av where A=12 43 6 12-7 (a) What is m? (b) What is n? (c) The image of T is a subspace of R. What is i? (d) The image of T is isomorphic to R. What is j? e The kermel of T is isomorphic to Rt. What is k7 (f) The kernel of T is a subspace of R. What is ?
Please put the solution in the form of a formal proof, Thank You. Let T: R3-R2 be the linear map given by a 2c (a) Find a basis of the range space. (Be sure to justify that it spans and is linearly independent.) (b) Find a basis of the null space. (Be sure to justify that it spans and is linearly independent.) (c) Use parts (a) and (b) to verify the rank-nullity theorem. Let T: R3-R2 be the linear map...
5 1 0 Problem 4: LetA = 0 41 . Consider the linear operator LA : R3 → R3 a) Find the characteristic polynomial for LA b) Let V-Null(A 51). V is an invariant subspace for LA. Pick a basis B for V and c) Let W-Null(A 51)2). W is an invariant subspace for LA Pick a basis a for W 0 3 2 use it to find LAlvls and the characteristic polynomial of LAl and use it to find...
Let T: R3 → R2 T(x, y, z) = (x + y,y+z) a. Is T a linear transformation? b. Find the matrix A of T C. Find the dimension of NUT and image T
Determine the image of the following map, and determine if it is linear. Let A = and take T: R4 ------- > R3 by T(x) = Ax