mid1_zpractice1_yPRACTICE EXAMINATION ONE FOR FIRST MID-TERM October 24, 2007 Math 21b, Fall 2007 MWF10 Evan Bullock MWF11 Leila Khatami MWF12 Yum-Tong Siu 1 20 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 10 10 10 Total 110 • Start by writing your name in the above box and check your section in the box to the left. • Try to answer each question on the same page as the question is asked. If needed, use the back or the next empty page for work. If you need additional paper, write your name on it. • Do not detach pages from this exam packet or un-staple the packet. • Please write neatly. Answers which are illegible for the grader can not be given credit. • No notes, books, calculators, computers, or other electronic aids can be allowed. • You have 90 minutes time to complete your work.Problem 1) TF questions (20 points) No justifications needed1)T FA linear system with 2 equations and 3 unknowns has either infinitely manyor no solutions.2)T FIf S is an invertible matrix which contains the vectors ~v1, ..., ~vnas columns,then ~v1, ..., ~vnis a basis of Rn.3)T FIf A, B are given n×n matrices, then the formula (A−B)(A+B) = A2−B2holds.4)T FSuppose A is an m × n matrix, where n < m. If the rank of A is m, thenthere is a vector y ∈ Rmfor which the system Ax = y has no solutions.5)T FThe matrix1 1 12 2 13 3 3is invertible.6)T FThe rank of an lower-triangular matrix equals the number of non-zero entriesalong the diagonal.7)T FThe row reduced echelon form of a 3 × 3 matrix of rank 2 is one of thefollowing1 0 ∗0 1 ∗0 0 0or1 ∗ 00 0 10 0 0.8)T FThe matrix"1 23 4#is a shear.9)T FFor any matrix A, one has dim(ker(A)) = dim(ker(rref(A))).10)T FIf ker(A) is included in im(A), then A is not invertible.11)T FThere exists an invertible 3 × 3 matrix, for which 7 of the 9 ent r ies are π.12)T FThe set of functions X = {f(x) = a sin(x)+b cos(x)+cx2+d | a, b, c, d ∈R}is a linear subspace of all continuous functions on the real line.13)T FIf A and B are n × n matrices, then AB is invertible if and only if both Aand B are invertible.14)T FThere exist matrices A, B such that A has rank 4 and B has rank 7 andAB has rank 5.15)T FThere exist matrices A, B such that A has rank 2 and B has rank 7 andAB has rank 1.16)T FIf f or a n invertible matrix A one has A2= A, then A = In.17)T FIf an invertible matrix A satisfies A2= I2, then A = I2or A = −I2.18)T FThe matrix"c − 1 −12 c + 1#is invertible for every real number c.19)T FFor 2 × 2 matrices A and B, if AB = 0, then either A = 0 or B = 0.20)T FIf T is a rotation in space with an angle π/6 around the z axes, then thelinear transformation S( x) = T (x) − x is invertible.2Problem 2) (10 points)Match each of matrices with one of the geometric descriptions below. You don’t have to giveexplanations.Matrix Enter A-H here.a)1 0 00 0 00 0 1b)0 −1 01 0 00 0 1c)−1 0 00 −1 00 0 −1d)1 0 00 1 00 0 −1Matrix Enter A-H here.e)1 0 00 1 00 1 1f)1 0 00 1 00 0 1g)1/2 1/2 01/2 1/2 00 0 0h)−1 0 00 1 00 0 −1A) Shear along a plane.B) Projection onto a plane.C) Rotation around an axes.D) Reflection at a point.E) Projection onto a line.F) Reflection at a plane.G) Reflection at a line.H) Identity transformation.Problem 3) (10 points)a) Write the matrix A ="1 −11 1#as a product of a rotation and a dilation.b) What is the length of the vector ~v = A100e1, where e1is the first basis vector?3c) In which direction does the vector ~v point?d) Find a matrix B such that B2= A.Problem 4) (10 points)Let A be a 3×3 matrix such that A2= 0. That is, the product of A with itself is the zero matrix.a) Verify that Im(A) is a subspace of ker(A).b) Can ran(A) = 2? If yes, give an example.c) Can ran(A) = 1? If yes, give an example.d) Can ran(A) = 0? If yes, give an example.Problem 5) (10 points)Let b, c be arbitrary numbers. Consider the matrix A =0 −1 b1 0 −c−b c 0.a) Find rref(A) and find a basis for the kernel and the image of A.b) For which b, c is the kernel one dimensional?c) Can the kernel be two dimensional?Problem 6) (10 points)Consider the matrix A =3 1 1 10 1 1 13 0 0 0.a) Use a series of elementary Gauss-Jordan row operatio ns to find the reduced row echelon formrref(A) of A. Do only one element ary operations at each step.4b) Find the rank of A.c) Find a basis for the image of A.d) Find a basis for the kernel of A.Problem 7) (10 points)Let A be a 2 × 2 matrix and S ="2 13 2#. We know that B = S−1AS ="1 10 1#. Find A2003.Hint. Write B = (I2+ C), note that C2= 0 and remember (1 + x)n= 1 + nx + .. + xn.Problem 8) (10 points)Let A be a 5 × 5 matrix. Suppose a finite number of elementary row operations reduces A tothe following matrix B =0 −1 0 0 00 0 0 0 1−1 0 0 1 10 −1 0 0 00 0 1 0 0.a) Find a basis of the kernel of A.b) Supp ose the elementary row operations used in reducing A to B are the following:i) Add row 2 to row 3.ii) Swap row 2 and row 4.iii) Multiple row 4 by 1/2.iv) Subtract row 1 from row 5.Find a ba sis of the image of A.Problem 9) (10 points)a) Find a basis for the plane x + 2y + z = 0 in R3.5b) F ind a 3 × 3 matrix which represents (with respect to the standard basis) a linear transfor-mation with image the plane x + 2y + z = 0 and with the kernel the line x = y = z.Problem 10) (10 points)Let T be the linear map from linear space X of 2 × 2 matrices to the real line which assigns tothe matrix A ="a bc d#its trace T (A) = a + d.a) (3 points) What is the dimension of the image of T ?b) (3 points) What is the dimension of the kernel of T ?c) (3 points) Find an explicit nonzero matrix in the kernel of T .d) (2 points) Is the transformation T (A) = det(A) = ad − bc a linear map from X
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