- Get Best Price Guarantee + 30% Extra Discount

- support@crazyforstudy.com
- +1-917-963-8942

- 797 step-by-step solutions
- Solved by professors & experts
- iOS, Android, & web

- Question : 1E - Let V be a vector space. Using the properties VS 1 through VS 8, show that if c is a number, then cO = O.
- Question : 2E - Let c be a number i= 0, and v an element of V. Prove that if cv = 0, then v = o.
- Question : 3E - In the vector space of functions, what is the function satisfying the condition VS2?
- Question : 4E - Let V be a vector space and v, W two elements of V. If v + W = 0, show that W= -v.
- Question : 5E - Let V be a vector space, and v, w two elements of V such that v + w = v. Show that w = O.
- Question : 6E - Let A 1, A2 be vectors in Rn. Show that the set of all vectors B in Rn such that B is perpendicular to both A 1 and A2 is a subspace.
- Question : 7E - Generalize Exercise 6, and prove: Let A 1,
- Question : 8E - Show that the following sets of elements in R 2 form subspaces. (a) The set of all (x, y) such that x = y. (b) The set of all (x, y) such that x - y = o. (c) The set of all (x, y) such that x + 4y = o.
- Question : 9E - Show that the following sets of elements in R 3 form subspaces. (a) The set of all (x, y, z) such that x + y + z = o. (b) The set of all (x, y, z) such that x = y and 2y = z. (c) The set of all (x, y, z) such that x + y = 3z.
- Question : 10E - If U, Ware subspaces of a vector space V, show that U n Wand U + Ware subspaces.
- Question : 11E - Let K be a subfield of a field L. Show that L is a vector space over K. In particular, C and R are vector spaces over Q.
- Question : 12E - Let K be the set of all numbers which can be written in the form a + b.j2, where a, b are rational numbers. Show that K is a field.
- Question : 13E - Let K be the set of all numbers which can be written in the form a + bi, where a, b are rational numbers. Show that K is a field.
- Question : 14E - Let c be a rational number> 0, and let y be a real number such that y2 = c. Show that the set of all numbers which can be written in the form a + by, where a, b are rational numbers, is a field.
- Question : 15E - Show that the following vectors are linearly independent (over C or R). (a) (1,1,1) and (0,1, -2) (b) (1,0) and (1,1) (c) (-1, 1,0) and (0, 1, 2) (d) (2, -1) and (1,0) (e) (n, 0) and (0,1) (f) (1,2) and (1, 3) (g) (1, 1, 0), (1, 1, 1), and (0, 1, -1) (h) (0, 1, 1), (0, 2, 1), and (1, 5, 3)
- Question : 16E - Express the given vector X as a linear combination of the given vectors A, B, and find the coordinates of X with respect to A, B. (a) X = (1,0), A = (1, 1), B = (0, 1) (b) X = (2,1), A = (1,-1), B = (1,1) (c) X = (1, 1), A = (2, 1), B = (-1,0) (d) X = (4,3), A = (2, 1), B = (-1,0)
- Question : 17E - Find the coordinates of the vector X with respect to the vectors A, B, C. (a) X = (1,0,0), A = (1, 1, 1), B = (-1, 1,0), C = (1,0, -1) (b) X = (1, 1, 1), A = (0, 1, -1), B = (1, 1,0), C = (1,0,2) (c) X = (0,0, 1), A = (1, 1, 1), B = (-1, 1,0), C = (1,0, -1)
- Question : 18E - Let (a, b) and (c, d) be two vectors in the plane. If ad - bc = 0, show that they are linearly dependent. If ad - bc # 0, show that they are linearly independent.
- Question : 19E - Consider the vector space of all functions of a variable t. Show that the following pairs of functions are linearly independent. (a) 1, t (b) t, t2 (c) t, t4 (d) et, t (e) tet, e2t (f) sin t, cos t (g) t, sin t (h) sin t, sin 2t (i) cos t, cos 3t
- Question : 20E - Consider the vector space of functions defined for t > O. Show that the following pairs of functons are linearly independent. (a) t, lit (b) e" log t
- Question : 21E - What are the coordinates of the function 3 sin t + 5 cos t = f(t) with respect to the basis {sin t, cos t}?
- Question : 22E - Let D be the derivative dldt. Let f(t) be as in Exercise 7. What are the coordinates of the function Df(t) with respect to the basis of Exercise 7?
- Question : 23E - Let A 1"" ,A, be vectors in Rn and assume that they are mutually perpendicular (i.e. any two of them are perpendicular), and that none of them is equal to O. Prove that they are linearly independent.
- Question : 24E - Let v, w be elements of a vector space and assume that v # O. If v, ware linearly dependent, show that there is a number a such that w = avo
- Question : 25E - Let V = R 2, and let W be the subspace generated by (2, 1). Let U be the subspace generated by (0, 1). Show that V is the direct sum of Wand U. If U' is the subspace generated by (1, 1), show that V is also the direct sum of Wand U'.
- Question : 26E - Let V = K3 for some field K. Let W be the subspace generated by (1, 0, 0), and let U be the subspace generated by (1, 1, 0) and (0, 1, 1). Show that V is the direct sum of Wand U.
- Question : 27E - Let A, B be two vectors in R2, and assume neither of them is O. If there is no number c such that cA = B, show that A, B form a basis of R2, and that R2 is a direct sum of the subspaces generated by A and B respectively.
- Question : 28E - Prove the last assertion of the section concerning the dimension of U x W If {u 1,

Just **$7.00/month**

Get immediate access to 24/7 Homework Help, step-by-step solutions, instant homework answer to over 40 million Textbook solution and Q/A

Pay $7.00/month for Better Grades

## Customer reviews

No reviews.