mid1_zpractice2_yPRACTICE EXAMINATION TWO 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 FIf A is invertible, then A3is invertible.2)T FIf A2is invertible, then A is invertible.3)T FThere is a matrix A and a vector~b such that A~x =~b has no solution andker(A) = {~0}.4)T FIf A is a 3 × 3 matrix which is a rotation around a line inR3then thecolumns of A form a basis ofR3.5)T FThe r ank of a diagonal matrix A equals the number of non-zero entries inA.6)T FRow reduction produces a diagonal matrix which has leading 1 at the placeswhere the original entries were not zero.7)T FA ="4/13 6/137/13 9/13#is a projection onto the line spanned by"23#.8)T FThe composition of a shear and a rotation in the plane is invertible.9)T FThe linear space of cubic polynomials ax3+ bx2+ cx + d has dimension 3.10)T FIf A is a 5 × 5 matrix such that A2= I5, (where I5is the identity matrix),then A is invertible.11)T FIf A is a 2 × 2 matrix such that A3= I2, then A is the identity matrix In.12)T FThere exists an invertible 3 × 3 matrix A such that 7 of its entries are 0.13)T FIt can happen that in rref(A) there are rows for which all entr ies arenonzero.14)T FIt is p ossible that rref(A) has columns for which all entries are nonzero.15)T FThe ra nk of a 3 × 4 matrix may be 4.16)T FThe ra nk of a 4 × 3 matrix may be 4.17)T FThere is an invertible 2 × 2 matrix S such that S−1"0 10 0#S ="1 00 0#18)T FThe number of leading 1 entries in rref(A) is the dimension of the image ofA.19)T FIf A and B are invertible n × n matrices, then so is A − B.20)T FThere exists a linear transformation T fromR3toR3for which ker(T ) =im(T ).Total2Problem 2) (10 points)Which of the following matrices are in reduced row-echelon form? We do no t need explanationsin this question.Matrix IS IS NOT Matrix IS IS NOTa)1 0 10 0 00 0 1b)1 0 10 1 00 0 1c)1 6 10 0 00 0 0d)1 6 00 0 00 0 1e)1 6 00 0 10 0 0f)0 1 10 0 10 0 0g)0 0 10 0 00 0 0h)0 0 00 0 00 0 0i)1 0 02 0 03 0 0j)0 0 10 1 01 0 0Problem 3) (10 points)In problems a)-b), you have to find all the solutions using Gauss-Jordan elimination.a) (4 points) Find all solutions of Axyz=001, whereA =1 1 01 2 52 3 4.b) (4 points) Find all solutions to the following system of linear equationsx + y + z + w = 4x − y + z − w = 232x + 2y + 2z + 2w = 8c) (2 points) You have a solution x ∈R9of a system of linear equations Ax = b, where A isa 7 × 9 matrix, and b is a given vector inR7. Is it po ssible to find an other solution Ay = b,where y is different from x?Problem 4) (10 points)Which of the following sets are linear subspaces of someRn? You do have to give explanations.1) (2 points) The image of the t ransformationR2toR2given by T ("xy#) ="1 −23 4#"xy#.2) (2 points) The kernel of the projection fromR3onto the xy-plane.3) (2 points) The solutions of the equation 2x + 3y − 5z = 12 inR3.4) (2 points) All the vectorsxyzinR3which satisfyhx y zixyz= 0.5) (2 points) All the points {~x ∈Rn| A~x = ~x }, where A is a given n × n matrix.Problem 5) (10 points)Let A be a shear in the plane along the x-axis which maps ~e1to ~e1and sends ~e2to 2~e1+ ~e2.One calls t his transformation also a horizontal shear. Let B the the projection onto the x-axis.a) (4 points) Find the matrices A, B, AB and BA.b) (3 points) What is A100?c) (3 po ints) What is (AB)10?Problem 6) (10 points)4a) (3 points) Let T be the linear transformation fro mR2toR2that is a reflection over thex-axis. Find the matrix of T .b) (4 points) Let S be the linear transformation fromR2toR2obtained by first reflecting overthe x-axis, then reflecting over the y-axis, and finally reflecting over the line y = x. Find thematrix A of S.c) (3 po ints) Find the inverse of A, where A is the matrix you found in part (b).Problem 7) (10 points)a) (3 points) Find a basis for the kernel ofA =h1 2 3 4 5ib) (3 points) Find a basis for the image of the following 4 × 6 matrixA =1 1 1 1 1 12 2 2 2 2 23 3 3 3 3 34 4 4 4 4 4.What is the dimension of the image and the dimension of the kernel?c) (4 po ints) Find the dimension of the image and kernel of the fo llowing 4 × 100 matrixA =1 2 3 · · · 99 1002 3 4 · · · 100 1013 4 5 · · · 101 1024 5 6 · · · 102 103.Problem 8) (10 points)Let ~v1="12#and ~v2="01#. Let B be the basis {~v1, ~v2} ofR2.a) (3 points) Find the coordinates of ~v ="56#in the basis B.In other words, find the B-coordinates of ~v.b) (4 point s) The matrix A ="1 13 4#defines a linear transformation T (~x) = A~x. What isthe B-matrix of T ?5c) (3 points) Is there a different basis B such that the B-matrix of T is"2 −21 −1#? Explainbriefly. (Here, T is t he linear transformation defined in part (b).)Problem 9) (10 points)You are given a matrix A which is a 5 × 6 matrix of rank 5. In each of the following questions,we need not only the answer but also a short explanation.a) (2 points) Can the matrix A be invertible ?b) (3 points) What is t he dimension of the image?c) (3 po ints) What is the dimension of the kernel?d) (2 points) How many solutions will the equation A~x =~b have?Problem 10) (10 points)A general shear is a linear t ransformation T for which there is a vector ~v with T (v) = v andsuch that for all vectors T (x) − x is parallel to v. Is the linear transformation T (x) = A(x)withA ="−1 4−1 3#a general shear? If so, find the line along …
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