Math Practice Problems for Test 1

Transcription

1 Math Practice Problems for Test UNSUBSTANTIATED ANSWERS MAY NOT RECEIVE CREDIT Show that w is in the span of v 5 and v 2 2 by writing w as a linear combination of v and v Find the value of k for the vector (, 2, k) R 3 to be a linear combination of u (3, 0, 2) and v (2,, 5). 3. Let A invertible. c 2. For what values of c is A not invertible? Find A 5 c + 2 when A is 4. Are the vectors w (,, 2, ), w 2 (, 2,, ) and w 3 (,, 4, ) linearly independent? Do they span R 4? 5. Verify that the set W {f C[0, ] : f(0) f()} is a vector subspace of C[0, ]. 6. Answer the following true or false. (a) T F {(0, 2a, a) R 3 : a R} is a vector subspace of R 3. (b) T F If three vectors v, v 2, v 3 span R 3, then they are linearly independent. (c) T F If three vectors v, v 2, v 3 are linearly independent, then they span R 3. (d) T F Any set of three vectors in R 3 is linearly independent. (e) T F The set {(a, b, c) R 3 : 2a b 3c 0} is a vector subspace of R 3. (f) T F Every linearly independent set of vectors in R 6 contains no more than six vectors. (g) T F Every set of vectors spanning R 4 contains at least 4 vectors. (h) T F The set {(x, y) R 2 : xy, 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). (j) T F The set {f C[, 2] : f() 3f(2)} is a vector subspace of C[, 2]. (k) T F Let A and B be n n matrices such that A exists. Then (ABA ) 2 AB 2 A.

2 (l) T F If A and B are matrices, and both AB and BA are defined, then A and B must be square matrices. 7. Let {v, v 2,..., v k } be a linearly dependent subset of a vector space V. Answer the following true or false: (a) T F v 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, v 2,..., v k is a linear combination of the others. (d) T F At least one of the vectors v, v 2,..., v k is a linear combination of the others. (e) T F dim(span{v, v 2,..., v k }) k. a b 8. Let A and B 2 c 2 0 (a) (7A) (b) (A T ) (c) A (d) (AB) (e) The solution to the equation B [ x x 2 ] 3 is 2 [ x x 2 ] 9. Consider the following the following system of linear equations: x + 2x 2 x 3 + 2x 4 + x 5 2 x 2x 2 + x 3 + 2x 4 + 3x 5 6 2x + 4x 2 3x 3 + 2x 4 3 3x 6x 2 + 2x 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. 0. Given the four points (20, 06), (30, 23), (40, 32), (50, 5): (a) Find the matrix representing the system of equations to find the coefficients of the cubic polynomial p(x) a 0 + a x + a 2 x 2 + a 3 x 3 that fits the four points above. (b) Find p(x).. Each month U-Haul trucks are driven among the cities Denver, St. Louis and Kansas City. Half of the trucks in Denver stay in Denver, with the remaining trucks split evenly between St.

3 Louis and Kansas City. /3 of the trucks in St. Louis remain in St. Louis while /6 of the trucks go to Kansas City and /2 of the trucks go to Denver. In Kansas City /5 of the trucks stay in Kansas City, /5 go to St. Louis and 3/5 go to Denver. (a) Let S Denver, S 2 St. Louis, and S 3 Kansas City. Below is the stochastic transition matrix P representing this system. Fill in the remaining entries of the matrix. 2 P 3 5 (b) If there are initially 200 trucks in Denver, 300 trucks in St. Louis, and 400 trucks in Kansas City, how many trucks are in Denver after one month? (c) How many trucks are in Denver after one year? 2. According to Hooke s law, when a force x is applied to a spring, the length of the spring y will be a linear function of the the force: that is, y a + bx, where a and b are the spring constants. Let various weights (in ounces) be suspended from a spring and the length of the spring measured in each case with results recorded in the table below. Force x Length y (a) The least squares regression model is given by what matrix equation? (b) Using (a), find the linear function that best fits the data points. y (c) Find the length of the spring if a force of 5 ounces is applied. 3. A simple economic system consists of the four industrial sectors of petroleum, textiles, transportation, and chemicals. One unit of petroleum requires.2 units of transportation,.4 units of chemicals and. unit of itself. One unit of textiles requires.6 unit s of petroleum,. unit of textiles,.5 units of transportations, and.3 units of chemicals. One unit of transportation requires.6 units of petroleum,. unit of itself, and.25 units of chemicals. One unit of chemicals requires.2 units of petroleum,. unit of textiles,.3 units of transportation, and.2 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 0 million dollars of chemicals. Find the output production for each sector to meet the external demands.

4 4. 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, x 2, x 3 and x 4. (b) Solve the system for x, x 2, x 3, x 4. (c) Find the traffic x, x 2, and x 3 when x Show that if A is an invertible matrix, its inverse is unique ( ) Answers - Practice Problems for Test 2. k 8 3. det A c 2 5 c + 2 (c 2)(c + 2) 5 c2 9. A is not invertible when c ±3. 4. They are linearly dependent as 3w + 2w 2 + w 3 (0, 0, 0, 0). They do not span R Verify that W {f C[0, ] : f(0) f()} is a subspace of C[0, ]. VSO: The constant function f defined by f(x) 0 has the property that f(0) 0 f(), therefore W. VS: Let f, g W. Then f(0) f() and g(0) g(). We want to show that the function f + g has this same property. (f + g)(0) f(0) + g(0) f() + g() (f + g)().

5 VS2: Let c R and f W. We want to show that the function cf has the right property to be an element of W. (cf)(0) c f(0) c f() (cf)() 6. (a) T (b) T (c) T (d) F (e) T (f) T (g) T (h) F (i) T (j) T (k) T (l) F 7. (a) F (b) T (c) F (d) T (e) F 8 (a) (7A) a b a 2 (b) (A 7 2 c T ) (A ) T b c (c) A (A ) c b ac + 2b [ 2 ] [ a ] 2 0 a b 2a 2b (d) (AB) B A [ ] [ ] 2 c a 2 b + c x (e) B x (a) (b) (c) x 2 s, x 5 t, x 2s + t 5, x 3 3, x 4 2 t (a) (b) p(x) x.3x +.003x (c) p(25) /2 /2 3/5 (a) P /4 /3 /5 /4 /6 / (b) P There are 490 trucks in Denver at the end of one month (c) P There are approximately trucks in Denver after one year

6 2. (a) A (X T X) X T Y where A a, X b 2 4 6, Y (b) y x (c) y(5) Petroleum 3. The input-output demand matrix is D Textiles and the Transportation Chemicals external demand matrix is E X DX + E. So, X (I D) E The output production of the petroleum sector is 20.7 million dollars, the output production of the textile sector is million dollars, the output production of the transportation sector is 07.3 million dollars, and the output production of the chemicals sector is million dollars. 4(a) 60 + x x x x 2 x x x 2 x rref (b) x 60 + t, x t, x t, x 4 t (c) x 20, x 2 30, x Show that if A is an invertible matrix, its inverse is unique. Proof: Suppose that AB BA I and AC CA I. We want to show that B C. Note that B BI B(AC) (BA)C IC C.