HW3 - AMS210.01. Homework 3 Andrei Antonenko Due at the...

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Homework 3 Andrei Antonenko Due at the beginning of the class, October 17, 2001 1. Which of the following statements are true or false? Explain your answer. If false, give a counterexample. (a) For any u 1 , u 2 , and u 3 from V , the system 0 ,u 1 ,u 2 ,u 3 is linearly dependent. (b) If u 1 , u 2 , and u 3 span V , then u 1 , u 2 , u 3 , and w span V for any w V . (c) If u 1 , u 2 , and u 3 span V , then dim V = 3. (d) If u 1 , u 2 , and u 3 span V , then they are independent. (e) If u 1 , u 2 , and u 3 form a basis for V , then they are independent. (f) If u 1 , u 2 are independent, then they form a basis of V . (g) If u 1 , u 2 , u 3 , and u 4 are linearly independent, then dim V = 4. (h) If u 1 , u 2 , and u 3 are linearly independent, then dim V 3. 2. Consider the vector space R 4 . Let u 1 = (1 , 2 , - 1 , 0), u 2 = (2 , 1 , 1 , 1), and u 3 = ( - 1 , 0 , 0 , 1). (a) Find 2 u 1 + u 2 - u 3 . (b) Find vector x R 4 such that 2 u 1 + u 2 + 2 u 3 + x = 0 . 3. Consider the vector space R 3 . Let u 1 = (1 , 2 , 1), u 2 = (2 , 1 , 3), and v = (1 , 2 ) , λ R . Find all λ ’s such that the vector v can be expressed as a linear combination of u i ’s. 4. Determine which of the following sets with standard operations form a vector space (over the field R ). Explain your answers. (a) { ( x,y,x ) | x,y R } (b) { (0 ,x,y ) | x,y R } (c) { ( x,y, 1) | x,y R } (d) { ( x,y,x + y ) | x,y R } (e) { ( x,y ) | x,y R ,x 0 } (f) Set of all polynomials of even power: ' f ( t ) = n i =0 a i t i | a i R ,n is even (g) Set of all symmetrical 2
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HW3 - AMS210.01. Homework 3 Andrei Antonenko Due at the...

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