10/22/2003 FIRST HOURLY PRACTICE Math 21b, Fall 2003Name:MWF10 Izzet CoskunMWF11 Oliver Knill• Start by writing your name in the above box andcheck your section in the box to the left.• Try to answer each question on the same page asthe question is asked. If needed, use the back orthe next empty page for work. (The actual examwill have more free space). If you need additionalpaper, 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 forthe grader can not be given credit.• No notes, books, calculators, computers, or otherelectronic aids can be allowed.• You have 90 minutes time to complete your work.1 202 103 104 105 106 107 108 109 10Total: 100Problem 1) (20 points) True or False? No justifications are needed.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.T FThe matrix1 1 12 2 13 3 3is invertible.T FThe rank of an lower-triangular matrix equals the number of non-zero entriesalong the diagonal.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.T FThe matrix"1 23 4#is a shear.T FFor any matrix A, one has dim(ker(A)) = dim(ker(rref(A))).T FIf ker(A) is included in im(A), then A is not invertible.T FThere exists an invertible 3 × 3 matrix, for which 7 of the 9 entries are π.T FThe dimension of the image of a matrix A is equal to the dimension of theimage of the matrix rref(A).T FThere exists an invertible n × n matrix whose inverse has rank n − 1.T FIf A and B are n × n matrices, then AB is invertible if and only if both Aand B are invertible.T FThere exist matrices A, B such that A has rank 4 and B has rank 7 andAB has rank 5.T FThere exist matrices A, B such that A has rank 2 and B has rank 7 andAB has rank 1.T FIf for an invertible matrix A one has A2= A, then A = 1.T FIf an invertible matrix A satisfies A2= 1, then A = 1 or A = −1.T FThe matrix"c − 1 −12 c + 1#is invertible for every real number c.T FFor 2 × 2 matrices A and B, if AB = 0, then either A = 0 or B = 0.T FThe determinant of a shear in the plane is always 1.T FThe plane x + y − z = 1 is a linear subspace of three dimensional space.T FIf T is a rotation in space around an angle π/6 around the z axes, then thelinear transformation S(x) = T (x) − x is invertible.TotalProblem 2) (10 points)Determine for each of the following matrices A, whether the system A~x = ~e1has zero, one orinfinitely many solutions and find the dimension of the image of A in each case:a)1 0 10 0 00 0 1.b)1 −2 02 0 00 0 3.c)5 0 00 2 20 2 2.d)1 4 50 1 30 0 −1.e)1 1 00 0 10 0 0.f)0 1 10 0 00 0 0.g)1 0 00 −1 00 0 −1.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?c) 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 zeromatrix.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 operations to find the reduced row echelon formrref(A) of A. Do only one elementary operations at each step.b) 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) Suppose 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 basis of the image of A.Problem 9) (10 points)a) Find a basis for the plane x + 2y + z = 0 in R3.b) Find 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 =
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