2. Suppose the linear operator L:R2 + R2 has matrix representation A = (Lee = (_}...
8. Find the standard matrix representation for each linear operator L: R2 + R2 described below: (a) L rotates each vector 7 by 45° in the clockwise direction. (b) L reflects each vector 7 about the 21 axis and then rotates it 90° in the counterclockwise direction. (c) L doubles the length of t and then rotates it 30° in the counterclockwise direction. (d) L reflects each vector 7 about the line x2 = 21 and projects it onto the...
With explanation! 3. Let B2 be the linear operator B2f (x):- f(0)2 2 (1f (1)2, which maps functions f defined at 0, 1 to the quadratic polynomials Pa. This is the Bernstein operator of degree 2, Let T = B21Py be the restriction of B2 to the quadratics. (a) Find the matrix representation of T with respect to the basis B = [1,2,2 (b) Find the matrix representation of T with respect to the basis C = (1-x)2, 22(1-2),X2]. (c)...
11. Suppose S: R R2 is the linear transformation with matrix -3 11 [2 -6 2 relative to the bases & and &. Find the matrix of S with respect to the bases (1,0, 1), (1,0,0), (1, 1,0)) and ((1,-1). (2,0). 11. Suppose S: R R2 is the linear transformation with matrix -3 11 [2 -6 2 relative to the bases & and &. Find the matrix of S with respect to the bases (1,0, 1), (1,0,0), (1, 1,0)) and...
AB matrix, linear operator R2 into R3 find the standard fro 11) For the linear operator L(x1, 22, ^1 + 22 AB matrix, linear operator R2 into R3 find the standard fro 11) For the linear operator L(x1, 22, ^1 + 22
11. =(7.5), #,(-3,-1) 2) Let = (1.-5). v. =(-2,2) and let L be a linear operator on R whose matrix representation with respect to the ordered basis . is a) Determine the transition matrix (change of basis matrix) from, v,to (1) (Draw the commutative triangle). 3 b) Find the matrix representation B, of L with respect to ,v} by USING the similarity relation
Let T be a linear operator on a finite dimensional vector space with a matrix representation A = 1 1 0 0] 16 3 2 1-3 -1 0 a. (3 pts) Find the characteristic polynomial for A. b. (3 pts) Find the eigenvalues of A. C. (2 pts) Find the dimension of each eigenspace of A. d. (2 pts) Using part (c), explain why the operator T is diagonalizable. e. (3 pts) Find a matrix P and diagonal matrix D...
36. Consider the linear operator T(x, y)- (5x-y,3x+2y) on R. Find the matrix of T with respect to the basis (4.3).(1,1) of R
9. Let T be a linear operator on R2 defined as follows on the standard basis of R2. T(1,0) = (3,2), T (0,1) = (-1,4). Find T(3, 5).
Linear CHALLENGE ACTIVITY 5.7.1: Matrix representation with respect to nonstandard bases. Jump to level 1 1 2 Let T : R3 + R2 be defined by T (6)-1 = 2x1 - 22 3x3 3 6 0 3 4 Let B uj = 7 , U2 , U3 2 and C= {v} = [:'], x==(-2]} What augmented matrix should be used to find (T]%, the matrix representation of T with respect to the bases B and C. Ex: 5 2 3...
only do (e)-(g) The linear operator L : R3 + R3 is given by its matrix A = Al,s wit respect to the standard basis S = {(1, 22, 23}, where To 0 11 -10- 20 [4 00 (a) Find the characteristic polynomial PL(x) of L; (b) What are the eigenvalues of L and what are their algebraic multiplicities? (e) What are the geometric multiplicities of eigenvalues of L? Is L diagonal- izable? (d) Find a basis B of eigenvectors...