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.
Let T be the linear transformation from R3 into R2 defined by (1) For the standard ordered bases a and ß for R3 and IR2 respectively, find the associated matrix for T with respect to the bases α and β. (2) Let α = {x1 , X2, X3) and β = {yı, ys), where x1 = (1,0,-1), x2 = - (1,0). Find the associated (1,1,1), хз-(1,0,0), and y,-(0, 1), Уг matrices T]g and T12
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|...
-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
Problem 2 [10pts] Let f : R3 + R2 be a linear transformation given by f((x, y, z) = (–2x + 2y +z, -x +2y). Find the matrix that corresponds to f with respect to the canonical bases of R3 and R2.
5. Let T: P2(R) + RP be the linear transformation that has the matrix …_…………..ນະ 1 2 -1 11 1 1 relative to the bases a = 1+ 21,1+1+12,1+for P2 (R) and B = (1,1),(1,-1) for R2. Find the matrix of T relative to the bases d' = 2+3.r,1+1+12,2+3.+r2 for P2(R) and B' =(3,-1),(1,-1) for R2.
Let T. R3 R3 be a linear transformation. Use the given information to find the nullity of T. rank(7) - 1 nullity(T) - Give a geometric description of the kernel and range of T. The kernel of T is a plane, and the range of T is a line. o The kernel of T is all of R3, and the range of T is all of R. The kernel of T is the single point {(0, 0, 0)), and the...
linear algebra Let T: R3 R3 be a linear transformation. Use the given information to find the nullity of T. rank(T) = 1 nullity(T) = Give a geometric description of the kernel and range of T. The kernel of T is the single point {(0, 0, 0)}, and the range of T is all of R3. O The kernel of T is all of R3, and the range of T is the single point {(0, 0, 0)}. The kernel of...
(1 point) Let f: R3 R3 be the linear transformation defined by f(3) = [ 2 1 1-4 -2 -57 -5 -4 7. 0 -2 Let B C = = {(2,1, -1),(-2,-2,1),(-1, -2, 1)}, {(-1,1,1),(1, -2, -1),(-1,3, 2)}, be two different bases for R. Find the matrix (fls for f relative to the basis B in the domain and C in the codomain. [] =
2. (5 points) Let T: R2 + R3 be a linear transformation with 2x1 - x2] 1-3x1 + x2 | 2x1 – 3x2 Find x = (x) <R? such that [0] -1 T(x) = (-4)
Q4. Let L: R2 + Rº be a transformation defined by L (0-2 [3u2 – U1 U1 – U2 -502 (a) Show that I is a linear transformation. (b) Find the standard matrix A of L, and find L ([31]) using the matrix A. (c) Do you think that any transformation T:R2 + R² is linear? (Justify your answer).