. (5 points) Given A [ 6 5 8 [ and b (a) Express the general solution of Ax b in parametric vector form. (b) Given that is a particular solution to Ax d, express the general solution to Ax d in parametric vector form.. (5 points) Use the matrix method to balance the chemical equation: 5.. (5 points) Let A (a) Find the inverse of A.. KClO KCl + O (b) What is (A T )? [ a + b abd b. ( points) Compute the determinant of c c + d ad bcd find it helpful to factor the entries wherever possible.) 5. (5 points) You are given the following matrix. A (a) Write an LU decomposition for A. 9 5 6 (b) Write the matrix L as a product of elementary matrices. 6. (9 points) Let A a e b f c g d h, R [ a b given that det c d, u a b c d and v 8. (You may Assuming that R is the reduced row echelon form of the matrix A, answer the following questions. (a) What are the vectors u and v? (b) Find a basis for Col(A). (c) How many vectors are in Col(A)? (d) Find a basis for Nul(A). e f g h. Page of 5 Question 6 continues on the next page.
(e) For what value(s) of k is 5 5 k (f) TRUE or FALSE: Nul(A T ) is a line. in Nul(AT )? 7. ( points) Assume that all matrices given below are n n and invertible, solve for the matrix X in B(X + A) C 8. (7 points) Let A be a matrix with det(a), and let I be the identity matrix. Furthermore, assume that A LU where L is unit lower triangular and U is upper triangular. Calculate: (a) det(l) (b) det(u) (c) det((a T ) A ) (d) det(la + A) 9. ( points) Suppose that A is an n n matrix. Show that if Nul(A) has dimension ero, then Nul(A ) must also have dimension ero.. ( points) Give an example of a non-invertible matrix A, for which det(a + I). { [ [ }. (5 points) Let H A M : A (a) Find a specific nonero matrix that is in H. (b) Given that H is a subspace, find a basis for it.. (6 points) Let V Span, and W Span,, (a) Show that if A is any matrix in V then A will be in W. (b) TRUE or FALSE: V is a -dimensional subspace of W. (c) TRUE or FALSE: W is a -dimensional subspace of V. w. (8 points) Let V x y : w and xy. (a) Is in V? (b) Find a nonero vector in V. Page of 5 Question continues on the next page.
(c) Is V closed under scalar multiplication? Justify your answer. (d) Is V closed under vector addition? Justify your answer. (e) Is V a subspace of R?. (6 points) Let T ABC denote the triangle whose vertices are the points A(, 6, 8), B(, 9, ), and C(, 6, 9). (a) Is the inner angle at the vertex B in T acute (between and π radians) or obtuse (between π and π radians). Explain your answer. (b) Find an equation of the form ax + by + c d for the plane through the point P (,, ) that is parallel to the plane containing the triangle T. x 5 5. (5 points) Let L denote the line given by the parametric vector equation y 8 + t, and let P denote the point (7,, ). Find the distance from L to P. 6. (5 points) Consider the line L in R given by x y + 8. x y 5 + t (a) Find the points on the line L that are unit away from the plane P. (b) Find the point where L and P intersect. and the plane P given by 7. (5 points) Let T : R R be the linear transformation that rotates vectors clockwise around the origin by θ, then reflects through the x axis, then rotates again by θ clockwise, and then reflects through the y axis. If T (x) Ax, find A. (Your final answer should not depend on the angle θ.) 8. (5 points) Let T : R R be a transformation such that ([ ) [ ([ ) [ 7 T, T ([, T (a) Based on the given conditions is T one-to-one? Explain your answer. [ [ [ (b) Express as a linear combination of and. (c) Is the transformation T linear? Justify. ) [ 7 9. ( points) Let T : U V be a linear transformation. Show that if T (u ) T (u ) then u u is in the kernel of T.. (7 points) Fill in the blanks with the word must, might, or cannot, as appropriate. (a) The non pivot columns of a matrix A form a linearly dependent set. Page of 5 Question continues on the next page.
(b) If A is an 5 8 matrix and rank (A) 5 then the linear transformation T (x) Ax be onto and be one-to-one. (c) If {a, b, c} is a linearly independent set in Span{u, v, w}, then {u, v, w} independent set. (d) The columns of an elementary matrix (e) If Col(A) Col(A T ) for a n n matrix A, then A form a linearly independent set. be a symmetric matrix. be a linearly (f) Given an n n matrix A. If the system Ax b is inconsistent for some b R n, then the system Ax have non-trivial solutions. Answers. (a) x (b) x + r + r. KClO KCl + O. (a) A 6. 8 5. (a) A 6. (a) u (d) 6 9,, v + s + s + t + t (b) (A T ) 5 6 (e) k (b) L (b) 6, (f) TRUE, 7. X C B A 8. (a) (b) (c) (d) det(l + I) det(a) 8 (c) Infinitely many Page of 5 Question continues on the next page.
9. dim(nul(a)) A is invertible and has det(a) det(a ) [det(a) A is also invertible [ and has dim(nul(a )). (many answers possible) [ {[ [ }. (a) (many answers possible) (b), (many answers possible). (a) A k + k +k k +(k +k ) A W (b) TRUE (c) FALSE (not a subset of V ) w. (a) Yes. (b) 8 (many answers possible) (c) Yes. If x y xy are true. So k (d) No. Counter-example: BA BC BA w x y 8, kw kx ky k V, then w and V since k(w) k() and (kx)(ky) k (xy) k ( ) (k) 8 V, but their sum 6 / V. (e) No. (b) x y +. (a) Since >, the inner angle at the vertex B must be acute. BC 5. 6 units 6. (a) (,, [ ) and (, 5 ) (b) (, 5, ) 7. A ([ ) ([ ) [ [ [ 8. (a) No. T and T yield the same result. (b) + ( [ [ ) ([ ) ([ ) 5 (c) No. T + +. 9. T (u u ) T (u ) T (u ) T (u ) T (u ) u u ker(t ). (a) MIGHT (b) MUST, CANNOT (c) MUST (d) MUST (e) MIGHT (f) MUST Page 5 of 5 Total: points