Consider the homogeneous system of linear equations 20 3r1 +2r2+40 Verify that the set of solutions...
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y...
(6) In R3, let W be the set of solutions of the homogeneous linear equation r + 2y +3z 0. Let L be the set of solutions of the inhomogeneous linear equation (a) Define affine subspace of a vector space. (b) Prove that L is an affine subspace of R3 (c) Compute a vector v such that L = v + W
Problem 5. Consider the following linear transformation. T12 11 + 2.13 3r1 +12 + 4.13 2r2+I3 L-2ri + 3.02 -.13 . Check if T is one-to-one. Then find ker(T). . Check if T is onto.
Question(9) (10 points) Consider the NON-homogeneous system of linear equations, AX b, given by: 01 2 3 89 10 113 (a) Is this set of equations consistent? (In other words, are there any solutions?) (b) If the equations are consistent, find ALL solutions x.
1. Consider the following system of linear equations: - 3x1 - 22 +2.03 = 7 2r2 - 2.23 = 8 6r1 - 312 + 6x3 = -9 (a) Put the system of linear equations into an augmented matrix. (b) Find the reduced row echelon form of the augmented matrix. (c) What is the rank of the coefficient matrix?
1. Verify that the following linear system does not have an infinite number of solutions for all constants b. 1 +39 - 13 = 1 2x + 2x2 = b 1 + bxg+bary = 1 2. Consider the matrices -=(: -1, -13). C-69--1--| 2 -1 0] 3 and F-10 1 1 [2 03 (a) Show that A, B, C, D and F are invertible matrices. (b) Solve the following equations for the unknown matrix X. (i) AXT = BC (ii)...
21 13 pts) 2. Find a basis for the solution space x of the following linear homogeneous system of equations: 1+2 +3 +14 213r2+4r3- 5x4 4x1+6r2 +8T3- 10x4 6r1 +9r2 +12r3 - 15r4= 0 0 Your solution must include verification that the basis spans the set of all solutions and is linearly independent.
21 13 pts) 2. Find a basis for the solution space x of the following linear homogeneous system of equations: 1+2 +3 +14 213r2+4r3- 5x4 4x1+6r2 +8T3-...
2. Consider the following system of linear equations 23 1 Determine whether this system is consistent, and if it is, find the full set of solutions. Also, find the rank of the matrix of coefficients.
2. Consider the following system of linear equations 23 1 Determine whether this system is consistent, and if it is, find the full set of solutions. Also, find the rank of the matrix of coefficients.
Suppose the solutions of a homogeneous system of five linear equations in six unknowns are all multiples of one nonzero so- lution. Will the system necessarily have a solution for every possible choice of constants on the right sides of the equations? Explain.
Is it possible that all solutions of a homogeneous system of twelve linear equations in fifteen variables are multiples of one fixed nonzero solution? Discuss. Consider the system as Ax = 0, where A is a 12 x 15 matrix. Choose the correct answer below. O A. No. Since A has 12 pivot positions, rank A = 12. By the Rank Theorem, dim Nul A = 12-rank A = 0. Since Nul A = 0, it is impossible to find...