Problem 4. Find basis and dimension of ker and im y for 9: R3 → R2...
Ler L: R4 → R3 be the linear transformation defined by (4p) L(z,y,z, t) = (x – y +t, 2x – 2, Y + 2z – t) a) Find the images of the standard basis of RA L(1,0,0,0) = L(0,1,0,0) = L(0,0,1,0) = L(0,0,0,1) = b) Find a basis and the dimension of the image of L c) Find a basis and the dimension of the kernel of L (8p) (8p)
2. a) Find the dimension of the solution space of the homogeneous linear system (1 point) x-3y + z = 0 2x-6y + 2z = 0 2x + 4y-82=0 b) Find a basis for the solution space. (1 point)
Find the basis and the dimension of the following linear
solution system:
x + y + z = 0, 3x + 2y – 2z = 0, 4x + 3y – z = 0 and 6x + 5y +z = 0
Define T : R3 → R2 by T(x,y,z) = (2x +4y +3z,6x) Show that T is linear.
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.
Let T: R3 → R2 T(x, y, z) = (x + y,y+z) a. Is T a linear transformation? b. Find the matrix A of T C. Find the dimension of NUT and image T
2. Consider the linear functions f: R3 → R2, 9: R3 R3, h: R2 + R and i: R3 → R4 given by: [ 5x – 72 1 * +54 +92 [2x + 3y +z] y = 3 +9y + 7z IL -+2y 2. + 2y + i i = y +22 |2y – z] (14 (a) Write them as matrices. (b) Which are the compositions we can do using two different functions from above? Do them using matrix multiplication.
2. Let Wi-((a, b, c) : a-c-b), W2-((a, b, c) : ab>0), W3-((z, y,z) : r2+92+22£1} be subsets of R3 (a) Determine which of these subsets is a subspace of R3. Justify your answer. (b) For the subsets which are subspaces, find a basis and the dimension for each of them
2. Let Wi-((a, b, c) : a-c-b), W2-((a, b, c) : ab>0), W3-((z, y,z) : r2+92+22£1} be subsets of R3 (a) Determine which of these subsets is a subspace...
What is the dimension of the subspace to a cubic
function
Problem 4 1. Write the solution set of 2r1 - 7x2 + lx3 = 9 in parametric vector form. Find a basis for the solutions in R3. What is the dimension of the subspace?
Problem 4 1. Write the solution set of 2r1 - 7x2 + lx3 = 9 in parametric vector form. Find a basis for the solutions in R3. What is the dimension of the subspace?
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.