Final Exam on Introduction to Linear Algebra (MAS109)
2012. 5. 24(Thu) 7:00 PM ∼ 10:00 PM
Spring 2012 MAS109
Full Name :
Student Number:
Class : A B C Do not write in this box.
Directions
• YOU MUST turn off your mobile phone. • No items other than pen, pencil, eraser and your ID Card are allowed. • Before you start, fill out the identification section on the title page and on the header of each page with an inerasable pen. • Show your work for full credit and write the final answer to each problem in the box provided. You can use the back of each page as scratch spaces. • Ask permission by raising your hand if you have any question or need to go to the toilet. • Any attempt to cheat in the exam leads to serious disciplinary actions not limited to failing the exam or the course.
Do not write in this table. Problem Score Problem Score Problem Score 1 /18 2 /20 3(a) /5
3(b) /5 3(c) /10 4(a) /5
4(b) /5 4(c) /5 4(d) /5
4(e) /5 5(a) /5 5(b) /10
5(c) /10 6(a) /10 6(b) /7
7(a) /7 7(b) /10 7(c) /10
8(a) /20 8(b) /8 Total /180
– 1 – POLICY FOR CHEATING
• Stealing a glance at others’ answer sheet or making noise: After warned once, you will be expelled from the room immediately; you will get scores of the problems answered up to that time. • Talking with others or using mobile phone in the room, corridor, or on your way to the toilet: Expelled immediately; you will get scores of the problems answered up to that time. • Making any memo on a sheet of paper or on a desk: Expelled immediately; NO score. • Proxy test takers: Proxy test taking is subject to a disciplinary action. NO score will be given, and you will be reported to the department. • Swapping the answer sheet (with another person or pre-written answer sheet): Answer-sheet-swapping is subject to a disciplinary action. NO score will be given, and you will be reported to the department.
I hereby confirm that I have read all the instructions written above, and I will follow the policy faithfully.
Signature:
– 2 – MAS109 Student Number: Name: 1. [3 points for correct answer, -3 points for incorrect answer, and 0 otherwise for each ques- tion.] Check if each of the following statements is true (T) or false (F). (a) For a set of vectors W ,(W ⊥)⊥ = W . (b) If A is an m × n matrix of rank k, then every row echelon form of A has m − k zero rows. (c) For an m × n matrix A, that Ax = 0 has only trivial solution means that A has full row rank. (d) Every idempotent matrix is symmetric. (e) If A is similar to B, and B is similar to C, then A is similar to C. (f) If A is an m × n matrix, then AA+ is the orthogonal projection of Rm onto row(A).
(a) (b) (c) (d) (e) (f) Ans: True False
– 3 – MAS109 Student Number: Name: 2. [20 pts] Let A be a 4 × 4 matrix given by 1 −1 2 3 −2 1 −1 1 . 4 −3 5 5 0 −1 3 7 . The reduced row echelon form of the partitioned matrix [A.I4] is given by 1 0 −1 −4 . 0 0 1/4 −3/4 0 1 −3 −7 . 0 0 0 −1 , 0 0 0 0 . 1 0 −1/4 −1/4 0 0 0 0 . 0 1 1/2 −1/2
where I4 is the identity matrix of size 4. Find bases for the four fundamental spaces of A.
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– 4 – MAS109 Student Number: Name: 3. For a matrix 1 1 5 3 [ ] . . . A = 1 2 6 2 = v1 . v2 . v3 . v4 2 5 13 3 (a) [5 pts] Find a subset of column vectors that forms a basis for col(A).
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(b) [5 pts] Express the other column vectors as a linear combinations of the basis vectors of question 3(a).
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– 5 – MAS109 Student Number: Name: (c) [10 pts] Find the Column-Row Factorization of A.
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4. [5 pts each] Consider the plane, W : x + 2y − 2z = 0.
(a) Find a unit vector u so that W = u⊥
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– 6 – MAS109 Student Number: Name: (b) Find a basis for W .
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3 (c) Find the matrix P for the orthogonal projection projW of R onto W .
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T (d) Find projW (x) where x = (1, 2, 0)
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– 7 – MAS109 Student Number: Name: (e) What is rank(P )?
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5. Consider the linear system Ax = b where 1 1 1 0 1 1 A = and b = 1 0 1 2 −2 1
(a) [5 pts] Compute rank(AT A).
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– 8 – MAS109 Student Number: Name: (b) [10 pts] Find the least squares solution of Ax = b.
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(c) [10 pts] Find the pseudoinverse A+ of A.
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– 9 – MAS109 Student Number: Name: 6. (a) [10 pts] Let A be the matrix 1 −1 6 A = 1 0 1 1 −2 1 Compute the QR decomposition of A.
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(b) [7 pts] Use the QR decomposition of A and solve the system Ax = b where b = (−1, 2, 0)T .
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– 10 – MAS109 Student Number: Name: { } ′ { ′ ′ ′ } 4 3 7. Consider the bases B = v1, v2, v3, v4 and B = v1, v2, v3 for R and R , respectively, T − − T − T T ′ in which v1 = (0, 1, 1, 1) , v2 = (2, 1, 1, 1) , v3 = (1, 4, 1, 2) , v4 = (6, 9, 4, 2) , v1 = T ′ − T ′ − T 4 → 3 (0, 8, 8) , v2 = ( 7, 8, 1) , v3 = ( 6, 9, 1) , and let T : R R be the linear transformation whose matrix with respect to B and B′ is 3 −2 1 0 [T ]B′,B = 1 6 2 1 . −3 0 7 1
(a) [7 pts] Find [T (vi)B′], i = 1, 2, 3, 4.
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(b) [10 pts] Find T (vi), i = 1, 2, 3, 4.
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– 11 – MAS109 Student Number: Name: 3 4 (c) [10 pts] Let S3 and S4 be the standard bases for R and R , respectively. Find a formula for
[T ]S3,S4. You don’t have to compute the formula. Just write down the full expression of the formula. For example, if the formula is given by PQ−1 and P and Q are matrices with some numeric values at the entries, then write down the full numeric expression of the matrices. You don’t have to compute the inverse of a matrix when the formula contains any inverse matrix.
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– 12 – MAS109 Student Number: Name: 8. Consider a linear operator T : R3 → R3 whose standard matrix is given by 3 9 −9 A = 2 0 0 3 3 −3
T T (a) [20 pts] Consider a basis B = {v1, v2, v3} with v1 = (0, 1, 1) , v2 = (a1, 0, a3) , and v3 = T 3 (b1, b2, 0) , for R . Let the matrix of T with respect to the basis B, i.e., [T ]B, is given by 0 1 0 0 0 1 . 0 0 0
Find a1, a3, b1, and b2.
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(b) [8 pts] Use the fact that A is nilpotent and compute exp(A).
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