LINEAR ALGEBRA AND VECTOR ANALYSIS MATH 22B

Transcription

1 3 4 Name: LINEAR ALGEBRA AND VECTOR ANALYSIS 8 9 MATH B Total : Unit 3: First Hourly Practice Welcome to the first hourly. It will take place on February 6 9 at 9: AM sharp in Hall D. You can already fill out your name in the box above. You only need this booklet and something to write. Please stow away any other material and electronic devices. Remember the honor code. Please write neatly and give details. We want to see details even if the answer should be obvious to you. Try to answer the question on the same page. There is also space on the back of each page. If you finish a problem somewhere else please indicate on the problem page so that we find it. You have 75 minutes for this hourly. Problems Problem 3P. ( points): We prove im(a) = im(aa T ) in two steps. Do them both. a) Prove that im(a) contains im(aa T ). b) Prove that im(aa T ) contains im(a). a) The image of AA T is contained in the image of A because we can write v = AA T x as v = Ay with y = A T x. b) On the other hand if v is in the image of A then v = Ax. If x = y + z where y in the kernel of A and z orthogonal to the kernel of A then Ax = Az. Because z is orthogonal to the kernel of A it is in the image of A T. Therefore z = A T u and v = Az = AA T u is in the image of AA T.

2 Linear Algebra and Vector Analysis Problem 3P. ( points): Decide in each case whether the set X is a linear space. If it is prove it if it is not then give a reason why it is not. a) The space of 4 4 matrices with zero trace. b) The space of 4 4 quaternion matrices. c) The space of matrices λq where Q is an orthogonal matrix and λ is real. d) The space of 3 3 matrices λq where Q is an orthogonal matrix and λ is real. e) The space of 5 5 matrices with entries or. f) The image im() of the identity 3 3 matrix. g) The kernel of the matrix A = [ 3 4]. h) The set of all vectors in R 3 for which x + y + z =. i) The set of all vectors in R 3 for which xyz =. j) The set of 3 3 matrices with orthogonal columns. a) Yes. Proof: is in X. Since tr(a + B) = tr(a) + tr(b) also the sum is in X. Since tr(λa) = λtr(a) also a scalar multiple of A is in X. b) Yes. you have even found a basis for it. a + ib + jc + kd where i j k are specific 4 4 matrices. The zero quaternion is the zero matrix. Quaternions can be added and scaled. [ ] [ ] c) No take A = and B = The sum is no more in the space. Note that if we would have taken for Q a rotation matrix (determinant ) then the answer would have been yes. It would have been the set of rotation-dilation matrices which form a linear space. d) No Take a similar counter example as in c like A = Diag( ) and B = Diag( ). e) No Take A = which is in the X but A = is not in X. (As a side remark it would be a linear space over the Boolean field Z = { } but for us we take real vector spaces unless stated otherwise.) f) yes The image of a matrix is always a linear space. g) Yes The kernel of a matrix is always a linear space. h) Yes This is the trivial space X = {}. Yes it is a linear space. i) No we can not add in this space like [ ] T + [ ] T = [ ] T is no more in the space. j) No take A = and B =. The sum A+B does not have orthogonal columns any more.

3 Problem 3P.3 ( points): a) ( points) Is the transformation T (x y) = (x y ) from R to R linear? b) ( points) Is the map T (A) = tr(a) as a map from M(3 3) to M(3 3) linear? Here is the identity matrix. c) (4 points) Match the transformation type: A: rotation dilation B: reflection dilation C: shear dilation D: projection dilation Fill A-D Matrix [ ] [ ] [ ] [ d) (4 points) SO SU or NO (no SO nor SU)? Yo you have to decide so! SOSUNO [ ] [ ] [ ] [ ] i i i i i i Matrix i i i i i ] a) No T (x y) T (x y). b) Yes. T () = T (x + y) = T (x) + T (y) and T (λx) = λt (x). c) Projection dilation rotation dilation reflection dilation shear dilation d) NO. This is i times a rotation dilation matrix. The determinant is not. NO. The determinant is zero NO. The determinant is zero SU. The determinant is. Problem 3P.4 ( points each sub problem is points): a) Which physicist promoted the use of examples to understand a theory? b) We describe a general linear map T from M( ) to M( ). In a suitable basis is this map given by a matrix or a 4 4 matrix? c) Is it true that f(x) = e 3x satisfies f(x) = O(e x )? d) Give an example of a 4 4 matrix A for which ker(a) = im(a). e) If B = S AS what can you say about the determinants of A and B? 3

4 Linear Algebra and Vector Analysis a) Feynman. b) 4 4 matrix. The space M( ) is equivalent to R 4. c) No. It is true that e O(3x) = e O(x) d) Take a partitioned matrix containing two magic matrix A in the diagonal implementing T (x y) = ( x). e) The determinants are the same. Problem 3P.5 ( points): In the checkers matrix the entry means that the checkers initial condition has a checker piece there and means that that field is empty: a) (6 points) Find a basis for the kernel of A. b) (4 points) Find a basis for the image of A. 4.

5 a) The kernel is 6 dimensional. To find it row reduce the matrix.. Introduce free variables a b c d e f and write down the equtions x = a c y = b d z = a y = b v = c w = d p = e q = f to get x y z u v w p q = a + b + c + d + e + f so that B ker (A) =. b) the image dimensional. It is spanned by the first two vectors in the checkers matrix because thats where the leading were in rref(a). B im (A) =. 5

6 Linear Algebra and Vector Analysis Problem 3P.6 ( points): a) (5 points) The projection-dilation matrix A = B = {v v v 3 } = { is given by a matrix B. Find this 3 3 matrix B. b) (5 points) A linear transformation T satisfies T (v ) = v T (v ) = v 3 T (v 3 ) = v in the basis where v v v 3 are given in a). Find the matrix R implementing this transformation in the standard basis. } a) b) B = S = B = S AS = R = SBS =. 6.

7 Problem 3P.7 ( points): Find the QR decomposition of the following matrices a)( points) A =. b) ( points) B = 3. 3 c) ( points)c = [ ] d) ( points)d =. [ ] e) ( points) E =. a) QR = b) QR = [ c) QR = A because A is already upper triangular d) QR = (A/ )( ). e) QR =. Both Q and R are the identity 7 ]. 5 3.

8 Linear Algebra and Vector Analysis Problem 3P.8 ( points): a) ( points) Find the determinant of the prime matrix 3 A = b) ( points) Find the determinant of the count to matrix B = c) ( points) Find the determinant of the - matrix C =. d) ( points) Find the determinant of the Pascal triangle matrix D = e) ( points) Find the determinant of the mystery matrix: E =

9 a) There is just one pattern with three upcrossing: = 365. b) Partitioning the matrix reveals two blocks. Thie determinant is the product of the determinants of the blocks. ( 6 5)(9 ) = 8. c). d) There is only one pattern. The number of upcrossings is so the determinant is. e) Row reduction. Subtract the first row from all others gives This matrix has determinant 5! =. 5 Problem 3P.9 ( points): Find the function which is the best fit for the data a x b x = y x y 3 - Write the data as a system of equations Ax = b with A = 3 3 and b = The [ rest is routine. ] We have x = [(A T A) A] T b = [ ] T /6. A few check points: (A T A) = The inverse is /56. A T b = [ 7] T. The best fit is y = (3/4) x (/4) x. 9.

10 Linear Algebra and Vector Analysis Problem 3P. ( points): The matrices A = C = i i i i B = D = are called the Gamma matrices or Dirac matrices. a) (4 points) Are the columns orthonormal? That is is it true that A A = where A = A T. Orthonormal columns? b) (3 points) Compute A and D and AD + DA. c) (3 points) Write down the inverse of AB and D. A B C D These were just routine computations. But they show something interesting about the anticommutation of the matrices. In physics there are written as γ j and the commutator is written as {A B} = AB + BA. What actually happens is that {γ j γ k } = g jk where g = diag( ) is the Lorentz metric. a) AB + BA = the zero matrix b) AD + DA = the zero matrix c) A = the identity matrix d) D = minus the identity matrix Oliver Knill knill@math.harvard.edu Math b Harvard College Spring 9