Practice Midterm Solutions, MATH 54, Linear Algebra and Differential Equations, Fall 4 Student ID: Circle your section: Shin 8am 7 Evans Lim pm 35 Etcheverry Cho 8am 75 Evans 3 Tanzer pm 35 Evans 3 Shin 9am 5 Latimer 4 Moody pm 8 Evans 4 Cho 9am 54 Sutardja Dai 5 Tanzer 3pm 6 Wheeler 5 Zhou am 54 Sutardja Dai 6 Moody 3pm 6 Evans 6 Theerakarn am 79 Stanley 7 Lim 8am 3 Hearst 7 Theerakarn am 79 Stanley 8 Moody 5pm 7 Evans 8 Zhou am 54 Sutardja Dai 9 Lee 5pm 3 Etcheverry 9 Wong pm 3 Evans Williams pm 89 Cory Tabrizian pm 9 Evans Williams 3pm 4 Barrows Wong pm 54 Sutardja Dai Williams pm Wheeler If none of the above, please explain: This is a closed book exam, no notes allowed. It consists of 6 problems, each worth points, of which you must complete 5. Choose one problem not to be graded by crossing it out in the box below. If you forget to cross out a problem, we will roll a die to choose one for you. Problem Maximum Score Your Score 3 4 5 6 Total Possible 5
Practice Midterm Solutions, MATH 54, Linear Algebra and Differential Equations, Fall 4 ) Decide if the following statements are ALWAYS TRUE (T) or SOMETIMES FALSE (F). You do not need to justify your answers. (Correct answers receive points, incorrect answers - points, blank answers points.) a) If v, v, v 3, v 4 are linearly independent vectors in R 6, then v + v, v 3 v 4 are linearly independent vectors. (T) If a(v + v ) + b(v 3 v 4 ) = then av + av + bv 3 bv 4 = so a = b =. b) The following linear system is inconsistent x + 4x 6x 3 + 8x 4 = x x + 3x 3 4x 4 = 5 (F) The system can be written as an augmented matrix and put in reduced row echelon form 4 6 8 3 4 5 3 4 5 There is no row of the form. c) If A is a 3 matrix and B is a 3 matrix, then the rank of the 3 3 matrix AB must be less than or equal to. (T) Since B is 3, we have rank(b), and so dim N ul(b). Hence dim N ul(ab), and so rank(ab). d) If two m n matrices A and B have the same reduced row echelon form, then they have the same column spaces. (F) Counterexample: A = B = e) det 3 4 5 (T) Row reduce to an upper triangular matrix: 3 4 = 4
Practice Midterm Solutions, MATH 54, Linear Algebra and Differential Equations, Fall 4 ) Circle all of the answers that satisfy the questions below. It is possible that any number of the answers (including none) satisfy the questions. (Complete solutions receive points, partial solutions points, but any incorrect circled answer leads to points.) a) Let A be an m n matrix. Which of the following is equal to m? Solution: v). i) rank(a) ii) dim Col(A) + dim Nul(A) iii) rank(a T ) iv) dim Col(A T ) dim Nul(A T ) v) dim Col(A T ) + dim Nul(A T ) b) Which of the following matrices is in reduced row echelon form? Solution: iii), v). i) ii) iii) iv) v) c) Which of the following conditions insures an m n matrix A is invertible? Solution: iv), v). i) m = n. ii) There exists an n m matrix B such that AB = I m. iii) The row echelon form of A has the same number of pivot rows as pivot columns. iv) Ax = b has a unique solution x for every b. v) A is injective and surjective. d) Which of the following T : R R is a linear transformation? Solution: ii), iv), v). i) T (x, y) = x + y + ii) T (x, y) = x y iii) T (x, y) = x + y (x + y) iv) T (x, y) = 6(x + ) + (y 3) v) T (x, y) = e) Suppose T : R 3 R 3 has -dimensional range and we know T (e ) = T (e 3 ) = Which of the following is a possible value of T (e )? Solution: i), iii), iv), v). 3 i) ii) iii) iv) v) 3
Practice Midterm Solutions, MATH 54, Linear Algebra and Differential Equations, Fall 4 3) Consider the matrix a) (5 points) Find bases for the column space and null space of A = Row reduce to find: N ul(a) basis: C ol(a) basis: b) (5 points) For what values of c is the vector v = c c c in the column space of A? Solve system: a + b = c c c Can do by row reduction, or observe that must have b = c, a = c and so c c = c. Thus need c = or. 4
Practice Midterm Solutions, MATH 54, Linear Algebra and Differential Equations, Fall 4 4) ( points) A linear transformation T : R R 3 satisfies the following: T ( ) = T ( ) = Find the standard matrix of T. We seek the 3 matrix with columns T (e ), T (e ). We have Thus we have T (e ) = T ((/) T (e ) = T ((/) = (/) = (/) + (/) = (/)T + ( /) = (/)T And so the matrix we seek is + (/) + (/) + ( /) ) = (/)T ( + ( /) = ) = (/)T ( / / / / = ) + (/)T ( / / ) + ( /)T ( / / ) ) 5
Practice Midterm Solutions, MATH 54, Linear Algebra and Differential Equations, Fall 4 5) Decide if each of the following matrices is invertible, and either find its inverse or justify why it is not invertible. a) (5 points) A = Row reduce to find inverse: b) (5 points) B = 4 4 4 4 Not invertible: column is twice column 4 so det B =. 6
Practice Midterm Solutions, MATH 54, Linear Algebra and Differential Equations, Fall 4 6) ( points) Suppose that v,..., v k are vectors in R n and that A is an m n matrix. Prove that if Av,..., Av k are linearly independent in R m, then v,..., v k are linearly independent. Suppose a v + a k v k =. We must show a = = a k =. Apply A to find A(a v + + a k v k ) =, and so A(a v ) + + A(a k v k ) =, and so a Av + + a k Av k =. Since Av,..., Av k are linearly independent, we have a = = a k =. 7