AB matrix, linear operator R2 into R3 find the standard fro 11) For the linear operator L(x1, 22,...
-00)0) 2 (AB 22) Let L : R, R2 be a linear transformation. You are given that L 2- 3 (a) Find the matrix A that represents L with respect to the basisu-| | 2-1 1-1 4 1 and the 6 standard basis F1 (b) Find the matrix B that represents IL with respect to the standard basis in both R3 and R2
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...
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...
R3 defined by 2. Let L be the linear operator on X21 X3X2 [x3- X1 L(x) and let S Span((1, 0, 1)) (a) Find the kernel of L (b) Determine L(S) (c) Determine L(R3). jes (d) Is L an onto mapping?
R2 defined as Consider the linear transformation T: R2 T(21,22)=(0,21 – 22) Find the standard matrix for T: a ab sin (a) f 8 ат What is the dimensi of ker(T)? Is T one-to-one? Enter one: yes no Write the standard matrix for HoT, where H is the reflection of R2 about the 3-axis. a sin(a) f 22 8 R a E är (Alt + A)
Consider the linear transformation T: R3 + R2 defined as T(X1, X2, 23)=(-23, -3 &1 – 23). Write the standard matrix for HoT, where H is the reflection of R2 about the y-axis. ab sin (a) a дх f a 12 ?
4. (12 pts) Show the matrix operator T: R3 R3 given by the following equations is one-to-one; Find the standard matrix for the inverse operator T-1, and find T-(W1, W2, W3). w1 = x1 +22-23 W2 = 2x1 +2:22 - 23 W3 = 21 - 2202
Let L : R2 → R3 be a linear transformation such that L 1 1 = 1 2 3 and L 1 2 = 2 1 3 . Find L 2 1 Find the standard matrix representing L. Find the dimensions of the kernel and the range of L and their bases. 12. Let L : R² + RP be a linear transformation such that L | (3) - -(5)-(1) Find I (*) Find the standard matrix representing L. Find...
6. Let L be the linear operator mapping R3 into R3 defined by L(x) Ax, where A=12 0-2 and let 0 0 Find the transition matrix V corresponding to a change of basis from i,V2. vs) to e,e,es(standard basis for R3), and use it to determine the matrix B representing L with respect to (vi, V2. V
3. (10 points) Let T : R3-A, T(x1, 22, 23) = (zi,-z2,23 a) Prove that T is a linear operator. b) Find the standard matrix of T.