Exercises 1. Consider the homogeneous equation 4x + 5y : 0 10 8 b. Prove that the general solution of the equation is (if) \"(:5)- c. Solve the vector equation x (45) : (lg). a. Prove that the following vector ( ) is a particular solution of this equation. 1.2 Vector subspaces and afne subspaces in R\" 21 2. Consider the inhomogeneous equation 7;: 3y + 2z 2 6. a. Prove that the following vectors are particular solutions of this equation. 0 1 5 2 0 , 1 , 9 , 4 3 2 1 16 b. Prove that the general solution of the equation is x 0 2 0 y = 0 +x 0 +y 2 z. 3 7 3 c. Represent each solution in part (a) in the form of the general solution. For example, 1 0 1 2 1 0 1 : 0 + 0 2 2 3 2 7 2 3 3. a. Let W be the vector subspace of solutions of the homogeneous linear equation 5x 8y : 0. Compute a set S of vectors in R2 such that W is the set of linear combinations of vectors in S. b. Let L be the afne subspace of solutions of the inhomogeneous linear equation SxSy: 1. Compute a vector v such that L : v + W. 4. a. Find all solutions of the homogeneous linear equation 3x2yz20. Write the set of solutions as a vector subspace. b. Find all solutions of the inhomogeneous linear equation 3x2yz=5. Write the set of solutions as an afne subspace. 22 1 Linear Equations and Vector Spaces 5. In R3, let W be the set of solutions of the homogeneous linear equation 2.x8y6z:0. Equivalently, x W: y eR3z2.xy6z:0 Z Compute a set S of vectors such that W = (S) is the set of linear combinations of vectors in S. 6. Consider the vector subspace x W: y 6R3z2xy6z20 z and and the afne subspace x L: y 6R3:2xy6z=l3 Z Compute a vector v such that L : v + W. '7. a. Find all solutions of the homogeneous linear equation 3x2yz20. Write the set of solutions as a vector subspace. b. Find all solutions of the inhomogeneous linear equation 3x2yz25. Write the set of solutions as an afne subspace. 8. Let W be the vector subspace of solutions of the homogeneous linear equation 5x3yz: 0. Compute a set S of vectors in R3 such that W = (S) is the set of linear combi- nations of vectors in S. Let L be the afne subspace of solutions of the inhomogeneous linear equation 5x3yz27. Compute a vector v such that L : V + W. . Let L be the set of solutions of the inhomogeneous linear equation x1+2x2+ 3x3 +4x4 : 15. 1.2 Vector subspaces and affine subspaces in R" 23 Write L as an affine subspace. 10. In the vector space R2, draw the vector subspace HER' : xty= 0 and the affine subspaces {(ERSixty= C) for c = -2, -1, 1, 2. 11. Consider the affine subspace L- { ( ) ER :xty= 1) Prove that ) EL if and only if ( * ) = (1-1)(6) + (9) for some t E R. 12. In the vector space R2, draw the vector subspace { ( ) ER2 : 5x - 24 = 0) and the affine subspaces { ( ) ER : 5x - 24= C} for c = -2, -1, 1, 2