Problem 1. q 1 1 _et a] = 4 and 2: 1 be two vectors in R3 . 3 o 131 1. Find all the vectors 5:: m2 iniiR3 that are orthogonal to 31 . Hint: Set up what it means to $3 be orthogonal; and you should get some system of equation that can be solved with \"week 1\" technology. m1 2. Find all the vectors 5:: m2 iniiR3 that are orthogonal to both a1 and db . 133 8. The set of vectors 5 you found in part 2 are really just all the vectors in the nullspace of some matrix :1 . What is this matrix x1 ? Problem 2. Consider the matrix A = 19 H' 1. Compute a basis for each of NS(A) , CS(A) , NS(AT) , and CS(AT) . 2. Plot the subspaces NS(A) and CS(A?) on the same xy -plane. You should observe that they are lines. What can say about those two lines? 3. Plot the subspaces NS(A") and CS(A) on the same xy -plane. You should observe that they are lines. What can say about those two lines?Problem 3. Consider the matrix A = 1 8 1/ . 1. Find a basis to NS(A) 2. Find a basis to CS(AT) 3. Show any vector ENS(A) is orthogonal to any vector ye CS(A") explicitly by writing a as some linear combination of the basis vectors you found in part 1, and writing y as some linear combination of the basis vectors you found in part 2, and take the dot product x y .Problem 4. Consider on the plane a square formed with vertices for some positive a >0 . Connect the diagonals, and form two vectors. Show these diagonals are perpendicular to each other using dot product