Math Some Exam Practice Problems Fall 7 Note that this is not a sample exam. This is much longer than your exam will be. However, the ideas and question types represented here (along with your homework) will help you prepare for your exam. All matrices in these questions should be assumed to have only real-valued entries. The exam covers material from text book sections.,.,.,.4,.5,.7,.8,.9,.. On the exam, you ll be allowed/given nothing except writing utensils. Unless otherwise stated, full credit always requires work and/or explanation. Specify any row operations that you perform. The following types of questions are all fair game: Questions like you ve seen in your homework Definitions and terminology questions True/False, Yes/No, short-answer, fill-in-the-blank questions Questions about anything in a blue or tan box in the text Questions that require written-word answers (might ask you to prove, show, explain, or discuss something).. Consider the following systems of equations. For each system, (i) Write the system as a matrix equation. (ii) Write the system as a vector equation. (iii) Give the augmented matrix of the system. (iv) Solve the system. Write the solutions in parametric form and in parametric vector form. x +6x +x = 5 x +7x x = 8 x x x = 5 5x 6x = 7 x x x =. Discuss whether the appropriate space (R, R, etc.) is spanned by each set of vectors. Identify which sets of vectors are linearly independent.,, 5 6 (d) [ ] [ ] [ ]},, 5 4 (e),, 7 [ ] [ ]}, 6, 5 4 (f) [ ] [ ]} 4, 9 6,. Determine all h and k, if any, so that the system has (i) no solution, (ii) a unique solution, (iii) many solutions. x +x = 4x +hx = 5 x + x = k x +hx = 5
4. For what values of h is v in Spanv,v }? For what values of h is v,v,v } linearly independent? 5 v =, v = 7, v = v =, v =, v = h 4 7 h 6 5. Determine the values of k such that the matrix is the augmented matrix of a consistent system. [ ] 4 6 k [ ] k 4 5 [ ] 7 4 k 4 6. Suppose A is a 7 4 matrix. How many pivots must A have if its columns are linearly independent? Why? Is it possible for the columns of A to span R 4? Why or why not? Is it possible for the columns if A to span R 7? Why or why not? 7. Suppose A is a 5 8 matrix. How many pivots must A have if its columns are linearly independent? Why? How many pivots must A have if its columns span R 5? Why? Is it possible for the columns of A to span R 8? Why or why not? 8. Let v, v, v, v 4, v 5 be vectors in R 5. Finish the following statement: By definition, v,v,v,v 4,v 5 } is a linearly dependent set if and only if Use the definition from to prove that if v =, then v,v,v,v 4,v 5 } is linearly dependent. Prove that if v Spanv,v,v 4,v 5 }, then v,v,v,v 4,v 5 } is linearly dependent. 9. Describe all solutions to Ax = in parametric form and in parametric vector form, where A is row equivalent to the given matrix. State whether the equation has only the trivial solution or infinitely many solutions. 7 4 5 5 5 6 7 6 4 6 8
. For each system, determine whether solutions exist. If solutions do exist, determine whether or not the system has a unique solution. Do as little reduction as possible. x +7x x x 4 +x 5 = 8 x x x + x 4 x 5 = 5 x +5x x 8x 4 = 8 x x x +4x 4 = x x +x +4x 4 = x 5x +x +8x 4 = 5 x + x 4x = x x x = 9 4x x +8x = 5 (d) x +x x 5x 4 = x x = x 4x 5x 4 = x x =. For those systems in problem that have solutions, describe their solution sets in parametric vector form. 4 5. Reduce the matrix 9 7 5 to echelon form, and then to reduced echelon form. 4 5. Find the general solution of the system whose augmented matrix is 6. 4. Determine if b is a linear combination of the columns of A. 5 A =, b = 4 A = 4, b = 4 4 5 6 4 5. Let v =, v =, v =. 5 Is v in Spanv,v }? If yes, express v as a linear combination of v and v. If no, explain why not. 6. For any v,v,v,v 4,v 5 R n, prove that v is in Spanv,v,v,v 4,v 5 }.
7. Compute the product Aw. [ ] 4 A =, w = A = [ ] [ ] 4, w = 4 8. Let v = 5, v = 4. List 5 vectors in Spanv,v }, showing the weights on v and v used to generate each vector. 9. Find a vector x whose image under T is b, where T is defined by T(x) = Ax. Determine whether x is unique. A =, b = 7 7. How many rows and columns must matrix A have so that the rule T(x) = Ax defines a transformation from: R into R 7? Explain. R 4 into R? Explain. 7/. Let A = 5. Find all x in R that are mapped to the zero vector by the transformation x Ax.. Let x = x x, v = x 4 5, w =, z = 4. Let T be a linear transformation that maps x to x v+x w+x z. Find a matrix A such that T(x) = Ax for every x in R. Find the image of u = under the transformation T.
. Let e =, e =, e =, and let T be a linear transformation such that T(e ) = x Find the images of 8 and x under T. x, T(e ) =, T(e ) = 6. 5 7 4 4. Let T : R R 6 be a linear transformation with T(e ) = (5,,,,,) and T(e ) = (,,,4,5,6), where e = (,) and e = (,). Find the matrix which implements T (find A such that T(x) = Ax for all x in R ). 5. Define transfomation T by T(x) = T(x,x,x ) = (x x 6x, x +x 4x,x x x, x +4x ). Does T map R onto R 4? Why or why not? Is T a one-to-one mapping? Why or why not? 6. Define transfomation T by T(x) = T(x,x,x ) = (x x 5x, x +x x,x x x ). Does T map R onto R? Why or why not? Is T a one-to-one mapping? Why or why not? 7. Construct a matrix A such that the columns of A do not span R, but the vector u = is in the 7 span of the columns of A. Explain how you built your A. 8. Suppose matrix A is m n. Discuss the relationship between the pivot positions in A and the number of solutions to Ax =.
9. a) Give the definition of a linear transformation. b) Complete this definition: A set of vectors v,v,...,v n } is linearly independent if and only if c) True or False: If A reduces to 4 d) True or False: For any matrix A, the equation Ax = is consistent., then the equation Ax = b is consistent for any b in R. e) Suppose T : R n R m is a linear transformation with standard matrix A. How many pivots must A have to ensure that T is an onto transformation?. f) If T(x)= solution? 7 8 x for all x R, is in the range of T? Does T(x) = have a non-trivial 4 5 g) Suppose matrix A is m n, with m > n. True or false: If A has n pivots, then Ax = b is consistent for each b R m. h) Suppose matrix A is m n, with m > n. True or false: If A has n pivots, then F(x) = Ax is a one-to-one transformation. i) Suppose matrix A is m n, with m < n. True or false: The transformation F(x) = Ax can not be an onto transformation. j) If matrix A is 5 5 and the product AB is 5 7, what is the size of B? k) Suppose the last column of AB is, but B itself has no column of all zeros. What can you say about the columns of A? l) True or false: If A is m n and B is n p, then AB has p columns.. For A = [ ], B = 4, I = [ ] [, u = ], compute the following: u T A u T u B( A I) (d) AB (e) AB T. Suppose matrix A is m n and B is n p. Show that if the columns of B are linearly dependent, then so are the columns of AB.