Math Practice Problems for Test 1


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1 Math Practice Problems for Test 1 UNSUBSTANTIATED ANSWERS MAY NOT RECEIVE CREDIT Let c 1 and c 2 be the columns of A 5 2 and b 1. Show that b Span{c 1, c 2 } by writing b as a linear combination of c 1 and c Find the value of k for the vector (1, 2, k) R 3 to be a linear combination of u (3, 0, 2) and v (2, 1, 5). 3. Let A invertible. c 2 1. For what values of c is A not invertible? Find A 5 c when A is 4. Are the vectors w 1 (1, 1, 2, 1), w 2 (1, 2, 1, 1) and w 3 (1, 1, 4, 1) linearly independent? Find a standard unit vector v R 4 such that v Span{w 1, w 2, w 3 } Show that combination.) Span( , , 0 0 ) (Hint: Write as a linear 6. Answer the following true or false. (a) T F {(0, 2a, a) R 3 : a R} is a vector subspace of R 3. a b (b) T F The 2 2 matrix is a symmetric matrix iff b c. c d (c) T F Any set of three vectors in R 3 span R 3. (d) T F Any set of three vectors in R 3 is linearly independent. (e) T F If a system of linear equations AX b is consistent, then A is invertible. (f) T F If P is a n n skewsymmetric matrix, then tr(p ) 0. (g) T F If a system of linear equations has two distinct solutions, then the system has three distinct solutions. (h) T F The set {(x, y) R 2 : xy 1, x, y R} is a vector subspace of R 2. (i) T F The set {A M 3,3 (R) : A A T } is a vector subspace of M 3,3 (R).
2 (j) T F If A is a n n matrix and c 1, c 2,, c n are the column vectors of A, then A is invertible iff {c 1, c 2,, c n } are linearly independent. (k) T F Let A and B be n n matrices such that A 1 exists. Then (ABA 1 ) 2 AB 2 A 1. (l) T F A homogeneous linear system of equations has an infinite number of solutions. 7. Let {v 1, v 2,..., v k } be a linearly dependent subset of a vector space V. Answer the following true or false: (a) T F v 1 Span{v 2, v 3,..., v k }. (b) T F The smaller set {v 2, v 3,..., v k } may still be linearly dependent. (c) T F None of the vectors v 1, v 2,..., v k is a linear combination of the others. (d) T F At least one of the vectors v 1, v 2,..., v k is a linear combination of the others. a b Let A 1 and B 2 c (a) (17A) 1 (b) (A T ) 1 (c) A (d) (AB) 1 (e) The solution set to the equation B x1 3 is 2 x1 9. Consider the following the following system of linear equations: x x 3 + 2x 4 + x 5 2 x x 3 + 2x 4 + 3x 5 6 2x x 3 + 2x 4 3 3x x 3 + 3x 5 9 (a) The augmented matrix representing this system is: (b) The reduced row echelon form of this matrix is: (c) Find all solutions to this system. 10. Find all 2 2 matrices A a 1 1 b such that A 2 I. a b 11. Given the four points (20, 106), (30, 123), (40, 132), (50, 151): (a) Find the matrix representing the system of equations to find the coefficients of the cubic polynomial p(x) a 0 + a 1 x + a 2 + a 3 x 3 that fits the four points above.
3 (b) Find p(x). (c) Find p(60). 12. Write a balanced equation for the following chemical reaction. C 2 H 6 O + O 2 CO 2 + H 2 O 13. A simple open economic system consists of the four industrial sectors of petroleum, textiles, transportation, and chemicals. $1.00 worth of output of petroleum requires $0.20 input of transportation, $0.40 input of chemicals and $0.10 input of itself. $1.00 worth of output of textiles requires $0.40 input of petroleum, $0.10 input of textiles, $0.15 input of transportations, and $0.30 input of chemicals. $1.00 worth of output of transportation requires $0.60 input of petroleum, $0.10 input of itself, and $0.25 input of chemicals. $1.00 worth of output of chemicals requires $0.20 units of petroleum, $0.10 unit of textiles, $0.30 units of transportation, and $0.20 units of itself. Units are measured in dollars. The external demands for the four sectors are 50 million dollars of petroleum, 220 million dollars of textiles, 330 million dollars of transportation, and 10 million dollars of chemicals. Find the output production for each sector to meet the external demands. 14. The flow of traffic (in vehicles per hour) through a network of streets is shown above. (a) Find the system of linear equations in terms of the variables x 1,, x 3 and x 4. (b) Solve the system for x 1,, x 3, x 4. (c) Find the traffic x 1,, and x 3 when x Answers  Practice Problems for Test 1
4 1. b ( 1)c 1 + 2c 2 2. k 8 3. det A c c + 2 (c 2)(c + 2) 5 c2 9. A is not invertible when c ±3. 4. They are linearly dependent as 3w 1 + 2w 2 + w 3 (0, 0, 0, 0). Note that (1, 0, 0, 0) Span{w 1, w 2, w 3 } (a) T (b) T (c) F (d) F (e) F (f) T (g) T (h) F (i) T (j) T (k) T (l) F 7. (a) F (b) T (c) F (d) T 8 (a) (17A) 1 1 a b a 2 (b) (A 17 2 c T ) 1 (A 1 ) T b c (c) A (A 1 ) 1 1 c b ac + 2b 2 a 2 0 a b 2a 2b (d) (AB) 1 B 1 A c a 2 b + c x (e) B (a) (b) (c) s, x 5 t, x 1 2s + t 5, x 3 3, x 4 2 t 2 a 1 a a + b b a + b b 2. So, a b (a) (b) p(x) x.31x+.003x3 (c) p(60) C 2 H 6 O + 3 O 2 2 CO H 2 O 13. The inputoutput demand matrix is C Petroleum Textiles Transportation Chemicals P etr T ex T rans Chem and
5 the external demand matrix is E X CX + E. So, X (I C) 1 E The output production of the petroleum sector is million dollars, the output production of the textile sector is million dollars, the output production of the transportation sector is million dollars, and the output production of the chemicals sector is million dollars x x x 14(a) x x rref x (b) x t, 80 + t, x t, x 4 t (c) x 1 210, 30, x 3 570
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