Problem 24 : Let b b2 b3 ba E R4 be a fixed vector, b + 0. Define L:R4 R by C1 12 L(x) = b. I, 2= ER4. 13 24 where bºx is the dot product of b and x in R4. (a) Show that L is a linear transformation. (b) Find the standard matrix representation of L. (c) Find a basis for and the dimension of Ker L. (d) Is L one-to one? Explain why. (e) Is L...
Problem 1: Let W = {p(t) € Pz : p'le) = 0}. We know from Problem 1, Section 4.3 and Problem 1, Section 4.6 that W is a subspace of P3. Let T:W+Pbe given by T(p(t)) = p' (t). It is easy to check that T is a linear transformation. (a) Find a basis for and the dimension of Range T. (b) Find Ker T, a basis for Ker T and dim KerT. (c) Is T one-to-one? Explain. (d) Is...
Let T: R4 → R3 be the linear transformation represented by T(x) = Ax, where 1 A = 0 -2 1 0 1 2 3 . 0 0 1 0 (a) Find the dimension of the domain. (b) Find the dimension of the range. (C) Find the dimension of the kernel. (d) Is T one-to-one? Explain. O T is one-to-one since the ker(T) = {0}. O T is one-to-one since the ker(T) = {0}. O T is not one-to-one since...
Let B = {b1,b2, b3} be a basis for a vector space V. Let T be a linear transformation from V to V whose matrix relative to B is [ 1 -1 0 1 [T]B = 2 -2 -1 . 10 -1 -3 1 Find T(-3b1 – b2 - b3) in terms of bı, b2, b3 .
solution of question d (4 points) Consider the basis of R5 given by with b2 (2,-1,-5,-4,7), b3-(3, 2,-7,-5,9) b4 2,1,4,4,-5) bs (-1,0,1,2,0) The MATLAB code to produce the basis vectors is given by b1 11,0-2-2.3], b2 -12-1.-5-4,7T, b3 13-2-7-5,91, b4 [-2,14.4-5T, b5 1-1,0,1,20 Let S denote the standard basis for R a Find the transition matrix P P,s PB,s b. Use the previous answer to calculate the coordinate matrix of the vector z ( 1,5, 4, 3, 3) with respect...
Question 2. Let 1 -15 B = 1 1 2 V2 a) Compute B2, B3, B4, B7, and B8. b) Use part a) to determine B2020. Show your work. c) The matrix B is invertible. Use part a) to find B-1. Justify your answer. (Note: no marks will be given if either the formula for the inverse of a 2 x 2 matrix or row reduction is used to compute B-1)
Problem 3 1. Prove that B (51, b2, b3,-4} {а, ег#3+ега, +6) is the basis for R4. al 2. Find 1 4 0 0 0 : 0 0 0 00 0 b 3. Consider the map T: R4-W with B-matrix B a 。), Find the standard matrix 1896 of T Problem 3 1. Prove that B (51, b2, b3,-4} {а, ег#3+ега, +6) is the basis for R4. al 2. Find 1 4 0 0 0 : 0 0 0 00...
Problem 13. Let l be the line in R' spanned by the vector u = 3 and let P:R -R be the projection onto line l. We have seen that projection onto a line is a linear transformation (also see page 218 example 3.59). a). Find the standard matrix representation of P by finding the images of the standard basis vectors e, e, and e, under the transformation P. b). Find the standard matrix representation of P by the second...
Question 5 For each given vector b and matrix A, determine if b e im(A) 1 -2 3 (a) b 0 A 21 3 0 5 15 (b) b A2-24 9 Question 6 Find the rank and nullity of the given linear transformations T and determine which are one-to-one and which are onto. r+ y ri+r2 Question 7 Find nullity(T) if (a) T:R R2, rank(T) 1 (b) T:RR, rank(T) 0 (c) T : Rs ? R2, rank(T)-1 Question 8 Let...
please provide detailed and clear solutions for the following 2-6 3 2- 0 -103-5 Calculate the determinants of A and B -1 4 (use either appropriate row and coumn expansions or elementary row operations and the properties of determinants). Are A and B invertible? Calculate their inverses if they exist 1b. Are the columns of A linearly dependent or linearly independent? Find the dimension of Nul A and the rank of A. What can you say about the number of...