MATH INFORMATION SHEET FOR TEST SPRING Test will be in PKH in class time, Tues April See above for date, time and location of Test It will last 7 minutes and is worth % of your course grade The material to be covered on the test is the same as that covered on the homework through the end of Chapter 6 inclusive ( Homework is at my website: http://wwwutaedu/math/vancliff/t/s/hwkhtml ) For my office hours, see my website In addition, I will be in my office (PKH 46) on Monday April from approximately : pm until 4: pm I will also have my usual office hour on the day of the test from approximately : pm until : pm in my office (PKH 46) This test will be mainly (but not entirely) multiple choice There will be several choices of answer per multiple-choice question and, for each, only one answer will be the correct one You should do rough work on the test or on paper provided by me No calculators, notes, cards, phones, tablets or computers are allowed BRING YOUR MYMAV ID CARD WITH YOU When I write a test, I look over the lecture notes and homework which have already been assigned, and the previous tests, and use them to model about 8% of the test problems (and most of them are fair game) You should expect 7- questions in total A good way to review is to go over the homework problems you have not already done & make sure you understand all the homework well by 48 hours prior to the test You should also go over the quiz and Tests & ; those tests and their solutions are posted online at my website (use password math to open the file); see http://wwwutaedu/math/vancliff/t/s/qspdf and http://wwwutaedu/math/vancliff/t/s/tspdf and http://wwwutaedu/math/vancliff/t/s/tspdf In addition, this information sheet provides some practice problems that are provided to help you study if you have finished all the homework questions These practice questions do NOT form a model for the test These questions are intended only to help you identify any gaps in your understanding In the last 4 hours before the test, reread ALL the homework problems, skim through the lecture notes, & go over these practice questions again Consider adopting one or more of the study techniques linked at my website Make sure you learn any formulae taught in class It is your responsibility to be on time Try to keep your eyes on your own work during the test All nontest items will have to be put by the side walls or by my desk prior to the start of the test; this includes cell phones & tablets, which need to be switched off & in with your belongings by the wall noncompliance will result in a score of on the test If you wish to leave the room during the test, you should ask permission first & turn in your test to me Only in exceptional circumstances will I let you continue the test should you return (So it is better to be minutes late to the test, rather than ask to go to the bathroom during the test) I plan to return the graded test to you on Thurs April or Tues April 8 Page of
The following questions are for your practice for the test to see how I might phrase some questions (especially multiple choice); they will not be graded and do NOT form a model for the test Finish the homework assignments first, before working them PRACTICE QUESTIONS Which of the following systems of equations is useful for determining the function(s) f(t) = a + bt + ct whose graph passes through the points (, ), (, 8), (, 8)? a = a + 4b = a b + c = 8 8 b = a + c = 8 a b + c = 8 a = a + 4b = 8 a b + c = 8 a = a + b + c = a b + c = 8 8 a = a + b + c = 8 a b + c = 8 a + b = a + c = 8 b c = 8 Which of the following systems of equations is useful for determining the function(s) f(t) = a cos(t)+b sin(t) that satisfy the equation for all t R? d f df + +f(t) = cos(t) dt dt 4a b = a + 4b = 4a + b = a 4b = a + 4b = a + 4b = a + 4b = a + b = a + b = a + b = a b = a b = Write down the coefficient matrix and the augmented matrix (stating which is which) for the x x x = system x x = x x = 4 The system of linear equations x + 4x + x = x + x = x + x = no solution exactly solution infinitely many solutions exactly solutions has Find the rank of 7 4 4 (g) (h) 6 Page of
6 Which one of the following explains why is not in reduced row-echelon form? Column contains a nonzero entry Column contains a pivot and another nonzero entry Column 4 contains a pivot and another nonzero entry Row contains a pivot and another nonzero entry Row contains a pivot and another nonzero entry The number of rows does not equal the number of columns (g) There is not a row of zeros in the matrix (h) Trick question: the matrix is in reduced row-echelon form [ ] 4 6 8 7 Find the reduced row-echelon form of 6 7 4 4 7 4 9 7 4 4 4 9 (g) (h) 4 6 8 If A = 8, then which of the following is a nontrivial (ie, nonzero) solution of the 6 9 equation Ax =? [ ] (g) (h) 9 If A = 4 8, then which of the following best describes the set of solutions to Ax = 7 9 geometrically? If The zero vector in R The zero vector in R a plane in R 4 a plane in R a line in R R (g) R 4 (h) points in R 4 k [ ] is the augmented matrix for a system of linear equations, then for which value(s) of k R is the system consistent? all k R except k = k = k = k = 4 all k R except k = all k R except k = 4 Justify or disprove: if a given vector b is a linear combination of the columns of a matrix A, then the equation Ax = b is consistent The product [ ] 4 7 equals Page of
[ ] + 7 + 4 [ ] 4 7 If 4 [ ] [ ] + 4 [ 7 [ 6 ] + ] 4 [ ] 8 [ ] 4 [ ] + 7 + trick question: this product is not defined is the augmented matrix for a system of linear equations, then the solution set of the system in vector form is: x = 4 + x x = 4 + x x = 4 + x there are no solutions not enough information given to answer question 4 A system of linear equations has augmented matrix where k R Find all values of k such that the system k, ( k)(k ) 6( k) does NOT have a unique solution all k [, ] ±, ± only,, 4 only 4 only only only (g), only (h) trick question: there is a unique solution for all k R Let A denote an m n matrix and let x R n Which of the following statements is always true? The system Ax = is always consistent The system Ax = is never consistent The system Ax = always has nonzero solutions If A is a matrix, then A must have a column without a pivot 6 Given that the reduced row-echelon form of is, k k 7 for which value(s) of k is a linear combination of and? k all k R except 7 k = only k = 7 only k = & k = 7 only all k R except & 7 all k R (g) there is no such k (h) not enough information given ( [ ] ) x x if x 7 Define T : R R x x by T = x [ ] if x = x Page 4 of
Show that there exist u, v R such that T (u + v) T (u) + T (v) (Hint: consider 8 If T : R R is a linear transformation such that T ( [ ] ) [ ] ( [ ] ) T =, find T 6 7 7 4 7 ( [ ] ) = [ ] [ ] 9 Let u, v R If T : R R is a linear transformation such that T (u) = find T (u v) 4 4 [ ] [ ] [ ] and [ ] [ ] [ ] and and T (v) = not enough information is given ( [ ] ) Find the matrix of the function T : R x R, where T = x [ ] x x [ ] [ ] x Suppose a line l in R contains the vector u = (orthogonal) projection map onto l 64 48 48 64 48 6 64 6 [ 48 ] 6 6 64 [ 64 ] 6 6 48 [ x ] [ ] 6 8 Find the shear matrix that transforms into 6 If A = x [ ] x of length [ 6 ] 48 48 64 no such matrix [ ] k, then for which value(s) of k is A not invertible? 4 & 4 Compute [ ] 4 4 [ ] 6 [ ] 6 Page of [ ] 7 4 [ ] 6 (g) cannot be multiplied no such matrix [ ] ) [ ], Find the matrix of the no such shear 4 7
If A =, then which of the following is the second row of A? 4 4 4 4 6 (g) (h) (i) A does not exist 7 6 Let v =, v = 6, v = Describe span(v, v, v ) = Rv + Rv + Rv 4 9 a point a line a plane R exactly points exactly vectors emerging from the origin in R 7 If A = 4 8, then which of the following best describes the kernel of A geometrically? 7 9 The zero vector in R The zero vector in R points in R a line in R a plane in R a plane in R 9 (g) R (h) R 4 8 If A = 4 8, then which of the following best describes the image of A geometrically? 7 9 The zero vector in R The zero vector in R points in R a line in R a plane in R a plane in R 9 (g) R (h) R 4 x 9 The set U = x : x, x R is not a subspace of R because U is not a subset of R U is empty U contains U contains the origin U does not contain the origin trick question: U is a subspace of R Page 6 of
9 Let v =, v =, v = Given that 7 is in the 4 9 kernel of the matrix A =, find a nontrivial relation among v, v and v 4 Show some work; write clearly Which of the following sets of vectors is linearly independent? 6,,,,, 6, 4 6 6 4, 4 7,, none of these 6 8 Which of the sets in the previous question is a basis for R? none Define (i) subspace of R n (ii) span of m vectors in R n (iii) kernel of a linear transformation (iv) image of a linear transformation (v) basis of a subspace (vi) dimension of a subspace (vii) the coordinate vector of a vector with respect to a basis B d 4 The set W = c b : a, b, c, d R and a + d = b + c is a subspace of R 4 Find the dimension a of W 7 6 4 (g) (h) You are given a matrix A and its reduced row-echelon form: 4 4 A = 7 4 6 rref(a) = 4 4 6 4 6 The matrix A represents a linear transformation T where T (x) = Ax and T : R R 6 T : R 6 R T : R 7 R 6 T : R 4 R 4 T : R 6 R 7 T : R 7 R (g) T : R R 7 (h) not enough information 6 Find the dimension of the image of the matrix A given in Question 6 4 (g) (h) not enough information Page 7 of
7 Find a basis for the image of the matrix A given in Question Rows,, 4 of rref(a) rows,, 4, of A columns,, 4, of rref(a) rows -4 of rref(a) all the rows of A all the columns of rref(a) (g) columns,, 4, of A (h) all the columns of A 8 Using the matrix A from Question and a vector x, how many free variables has the homogeneous system Ax =? 7 6 4 (g) (h) 9 Find the dimension of the kernel of the matrix A given in Question 7 6 4 (g) (h) 4 Find a basis for the kernel of the matrix A given in Question ; show all work 4 Let A = [ ] [ ], v 8 = [ ] 4 [ ], v = [ ] [ ] Using B = {v, v } as a basis for R, find [Av ] B [ ] [ ] not enough information 4 Let T (x) = Ax for all x R, where A is the matrix from Question 4 Find the matrix of T with respect to the basis B given in Question 4 [ ] 8 8 8 8 [ ] (g) [ ] (h) [ ] a a 4 Find a basis for the real linear space V = b b R : a, b, c R c c,,,,,,,,, (g) V has a basis, but it is infinite 44 Let A, B R n n We define a map T : R n n R n n by T (M) = AMB for all M R n n Justify whether or not T is a linear transformation Show all work Page 8 of
[ ] 4f() 4 Let T : P R be given by T =, where f denotes the derivative of the polynomial f; ie, T = df [ f (4) ] 4f(t) t= You may assume that T is a linear transformation Using the dt t=4 {} basis {t, t, } for P and the basis, for R, find the matrix of T with respect to these bases [ ] 4 4 4 8 4 [ 4 4 ] 4 8 4 8 4 4 4 4 4 8 46 Using the basis B = {t, t, t, } for P, find [t + t ] B [ 4 4 ] 4 8 4 4 4 8 4 (g) 47 Let V be the plane, V = Rv + Rv in R, where v =, v = (h) Let T : V V be a linear transformation such that T (v ) = v v and T (v ) = v Find the matrix of T with respect to the basis {v, v } for V [ ] (g) 48 Let e =, e = and e =, and let E = {e, e, e } be the standard basis of R, and assume B = { e, e, e + e } is another basis of R Find the matrix which changes basis from B to E (g) (h) 49 Let u, u, u denote orthonormal vectors in R Find the length of the vector 4u u + u (g) (h) Page 9 of
Which of the following sets of vectors form an orthonormal set?,,,,,,,,,, none of them Use the Gram-Schmidt orthogonalization procedure to produce an orthonormal basis for the subspace of R spanned by and,,,,, (g), (h), Which one of the following matrices is orthogonal? [ ] 4 4 [ ] (g) [ ] Which one of the following matrices is symmetric? [ ] 8 8 9 6 [ ] [ ] [ ] 4 Find C T + AB, where A =, B =, C = 4 9 4 4 4 4 4 4 4 4 4 4 (g) (h) (i) cannot be computed 4 7 Find the determinant of (g) (h) 7 (i) 9 Page of (g) [ ]
6 If A, B and C are n n matrices such that det(a) =, det(b) =, det(c) = 4, find det(ab C T ) 4 4 6 (g) 8 (h) (i) 7 If A is a matrix, find det(a) 8 det(a) det(a) 8 det(a) det(a) det(a) (g) 8 det(a) (h) det(a) (i) 8 det(a) I plan to post the answers to the multiple-choice problems herein at my homework website ( http://wwwutaedu/math/vancliff/t/s/hwkhtml ) at roughly : pm Mon April, so work the problems first and then compare with the posted answers You might find it beneficial to pick 4 of these questions and answer them in a simulated test environment ; ie, time yourself for minutes and do not use your notes or book etc This will allow your brain/mind practice what it will be like to answer Math questions in a test environment Remember: most of the test will be based on the homework questions, so finish those questions and look over your solutions to them before the test Reread Page above to see what to bring to the test Page of