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1. Consider the transformation given by T(x, y, z)- (2z 3z+) (a) Show that T is...
12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal. 12: Find a basis B for R', such that the matrix for the linear transformation T: R' R', T(x,y,z)-(2x-2z,2y-2z,3x-3z) relative to B is diagonal.
Show your work! No work, no credit! 1. Given T[c x, y, z >)-< x-z, y >. Complete the following a. Check if T is a linear transformation. Show your work! b. Find the domain and range of T. c. If T is a linear transformarion, find the matrix A that induced T. (6 points) 1)
Find the standard matrix for the linear transformation T. T(x, y, z) = (x - 2z, 2y = z) 11
Consider the matrix transformation T:R → R given by T(x,y,z) = (x+ay, x+(a+1)y, x+ay+z) where a = 13. First use inverse of transformation to find T-(2,1,2). if T-(2,1,2)=(b,c,d) then b+c+d =
2. [& marks] Consider the line ar transformation T: R – R? T(x,y,z) = (x +y-2, -1-y+z). (a) Show that the matrix [T]s, representing T in the standard bases of Rand R' is of the form [7|6,6= ( +1 -1 1). -1 -1 1 (b) Find a basis of the null space of T and determine the dimension of this space. (c) Find a basis of the range of T and determine the dimension of the range of T. (d)...
2. (8 marks] Consider the linear transformation T:R3 R2 TX,Y, 2) = (+y-2, -1-y+z). (a) Show that the matrix (TE.Es representing T in the standard bases of R3 and R² is of the form TEE 1 -1 1 -1 -1 1 (b) Find a basis of the null space of T and determine the dimension of this space. (c) Find a basis of the range of T and determine the dimension of the range of T. (d) Is T Onto?...
20. Consider the transformation from R →Rdefined by T(x, y, z) = (x + y, z). a. Under this transformation, find the image of the ordered pair (1, -3, 2). b. Is the transformation linear? Show your work! [5 marks]
(1 point a. The linear transformation T : R2 → R2 is given by: Ti (x, y) = (2x + 9y, 4x + 19y). Find T1x, y). 「-i(x, y) =( x+ y, x+ b. The linear transformation T2 : R' → R' is given by: T2(x, y, z) (x + 2z,2r +y, 2y +z) Find (x, y, z). T2-1(x,y,z)=( x+ y+ z, x+ y+ z, x+ y+ z)
= Let T:R3 → Rº be the linear transformation given by T(x,y,z) = (x – 2, x + y, x + y + 2z) for all (x,y,z) e R3. Determine whether T is invertible or not. If T is invertible, find the inverse of T and compute inverse image of (1,1,1) under T.
Find the augmented matrix of the linear system X +y+z= -8 X – 3y + 3z = -4 X – Y + 2z = -6. Use Gauss-Jordon elimination to transform the augmented matrix to its reduced row- echelon form. Then find the solution or the solution set of the linear system.