Linear Algebra, Fall 2007 (http:/www.math.nthu.edu.tw/ wangwc/)
Study Guide for Chapter 2
1. A typo in Study guide for Chap 1: Section 1.3(b), Theorem 1.3.1 should be 1.3.2 instead. 2. Section 2.1: (a) Know how to compute determinant inductively (Theorem
80
Chapter 5
17. (a) A is symmetric since AT = (xyT + yxT )T = (xyT )T + (yxT )T = (yT )T xT + (xT )T yT = yxT + xyT = A (b) For any vector z in Rn Az = xyT z + yxT z = c1 x + c2y where c1 = yT z and c2 = xT z. If z is in N (A) then 0 = Az = c1 x + c2y an
Chapter 5
SECTION 1
14 1. (c) cos = , 10.65 221 46 (d) cos = , 62.19 21 3. (b) p = (4, 4)T , x p = (1, 1)T pT (x p) = 4 + 4 = 0 (d) p = (2, 4, 2)T , x p = (4, 1, 2)T pT (x p) = 8 + 4 + 4 = 0 4. If x and y are linearly independent and is the angle between
Chapter 4
SECTION 1
2. x1 = r cos , x2 = r sin where r = (x2 + x2)1/2 and is the angle between 1 2 x and e1 . L(x) = (r cos cos r sin sin , r cos sin + r sin cos )T = (r cos( + ), r sin( + )T The linear transformation L has the eect of rotating a vector b
48
Chapter 3
SECTION 4
3. (a) Since 2 1 4 3 =2=0
it follows that x1 and x2 are linearly independent and hence form a basis for R2. (b) It follows from Theorem 3.4.1 that any set of more than two vectors in R2 must be linearly dependent. 5. (a) Since 2 3 2
CHAPTER 3
SECTION 1
3. To show that C is a vector space we must show that all eight axioms are satised. A1. (a + bi) + (c + di) = (a + c) + (b + d)i = (c + a) + (d + b)i = (c + di) + (a + bi) A2. (x + y) + z = [(x1 + x2i) + (y1 + y2 i)] + (z1 + z2 i) = (x
14
CHAPTER 1
Conversely suppose there exists a nonsingular matrix M such that B = M A. Since M is nonsingular it is row equivalent to I . Thus there exist elementary matrices E1 , E2, . . . , Ek such that M = Ek Ek1 E1I It follows that B = M A = Ek Ek1 E1
Chapter 6
SECTION 1
2. If A is triangular then A aiiI will be a triangular matrix with a zero entry in the (i, i) position. Since the determinant of a triangular matrix is the product of its diagonal elements it follows that det(A aiiI ) = 0 Thus the eige
Linear Algebra, Fall 2002
Final Exam
Jan 10 2003, 13:10 PM - 15:00 PM. SHOW ALL YOUR WORK
(1) (10 pts) Let S = spancfw_(1, 2, 3)T , nd a basis for S . (2) (20 pts) Given a set of data (x, y ) = cfw_(1, 0), (0, 1), (1, 3), (2, 5) (a) Find the best least sq
Linear Algebra, Fall 2007 (http:/www.math.nthu.edu.tw/ wangwc/)
Study Guide for Midterm 1
1. Review the homework problems. 2. Check out the study guides. Those marked skipped will not be examined. 3. Check out the Chapter Test at the end of Chapter 1 and
Linear Algebra, Fall 2007 (http:/www.math.nthu.edu.tw/ wangwc/)
Homework Assignment for Week 16
Assigned Dec 28, 2007 1. Section 6.1: Problems: 17, 18, 20, 27. Hint for Problem 18: Here | denotes the length of the (possibly) complex eigenvalue x1 . . If =
Linear Algebra, Fall 2007 (http:/www.math.nthu.edu.tw/ wangwc/)
Homework Assignment for Week 15
Assigned Dec 21, 2007 1. Section 5.6: Skip Modied Gram-Schmidt Process. 2. Section 5.6: Read and Example 2 for a simpler way of computing the matrix R in QR fa
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Linear Algebra, Fall 2002
Quiz 2
Nov 29 2002, 13:10 PM - 14:25 PM
(1) (15 pts) Which of the following are subspaces of R2 ? Explain. (a) y = x (b) y = x2 (c) y = 2x + 1 (2) (20 pts) True of False? Explain. If S and T are subspaces of a vector space V , so
Linear Algebra, Fall 2002
Midterm 1
Nov 01 2002, 13:10 PM - 15:00 PM
(1) (10 pts) 121 A = 1 1 0 111 Compute adjA (2) (15 pts) A Mm,n , A Mn,p . Is it true that (AB )T = B T AT ? Explain. (3) (20 pts) How many terms are there in the determinant of A M5,5 ?
f f f f f f f f f f ef 873jjihh1j2887766670iiiiiihggggggjh8e888887777777jjjjjjjjjjkkkkkkkiiiiiiiiiiiiiiiiihhhhhhhhhhhhhhhhhgggggggggggggggggfd 1e fd ge fd ie fd e fd jif fd jd kd fe jd 1d fe jd 7d fe jd d iiii fe fe id jd kd 3d ggggggfe fe h0d id jd kd 7d
Linear Algebra, Fall 2007 (http:/www.math.nthu.edu.tw/ wangwc/)
Study Guide for Chapter 6 1. Section 6.1: (a) Study how to nd the eigenvalues and eigenvectors of a square matrix. (b) Remember an example of a real matrix with complex eigenvalues and eigenv
Linear Algebra, Fall 2007 (http:/www.math.nthu.edu.tw/ wangwc/)
Study Guide for Chapter 5
1. Section 5.1: (a) Pay attention to scalar and vector projection of x onto y . (b) Skip Application 1,3 . Read Application 2. 2. Section 5.2: (a) Study the proof of
Linear Algebra, Fall 2007 (http:/www.math.nthu.edu.tw/ wangwc/)
Study Guide for Chapter 4
1. Section 4.1: (a) Understand and and memorize the matrix representations of reections, projections, dilations and rotations. (b) Study examples on linear transform
Linear Algebra, Fall 2007 (http:/www.math.nthu.edu.tw/ wangwc/)
Study Guide for Chapter 3
1. Section 3.1: (a) Do not try to memorize A1-A8. (b) What are the closure properties? (c) Memorize a few examples of vector space other than Rn . 2. Section 3.2: (a