-mtrxthequiz2_1011

# -mtrxthequiz2_1011 - 8 Set U = a b c | abc = 6 a,b,c ∈ R...

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MTRXTHE QUIZ 2 November 3, 2010 Direction : Do as indicated. SHOW YOUR COMPLETE SOLUTION. 1. Use Cramer’s Rule to solve the system of equations. (8 Points ) x + 2 y - z = - 2 2 x + y + z = 0 3 x - y + 5 z = 1 2. Determine if the following subsets are subspaces of V . (10 Points ) (a) S = a b c , a 0 , b 0 , c 0 , V = R 3 (b) S = a + 3 b a - b 2 a + b 4 a , a, b R , V = R 3 3. Let S = { v 1 , v 2 , v 3 } where v 1 = 1 0 - 1 , v 2 = 2 1 3 , v 3 = 4 2 6 . (12 Points ) a. Does S span R 3 ? b. Is S linearly independent? c. Is S a basis for R 3 ? 4. Show that S = { 1 , 2 t, - 2 + 4 t 2 } is a basis for P 2 and express 3 - t + 2 t 2 as a linear combination of this basis. (10 Points ) 5. Define “real vector space”. Give an example other than the set of real numbers or R n or R n . (10 Points ) 6. Define “linearly independent”. Give an example in R 2 . (5 Points ) 7. Determine if the vector ( - 2 , 0 , 3), can be expressed as a linear combination of the vectors (1 , 3 , 0) and (2 , 4 , - 1). (5 Points
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Unformatted text preview: ) 8. Set U = { ( a b c ) | abc = 6; a,b,c ∈ R } , a subset of R 3 . Give three vectors from U , and determine whether or not U is a vector space. (10 Points ) 9. Determine if the set of all 3 × 3 diagonal matrices is a vector space under the usual matrix addition and scalar multiplication. (10 Points ) BONUS. Let F [ a,b ] be the set of all real valued functions that are deﬁned on the interval [ a,b ]. Then given any two “vectors”, f = f ( x ) and g = g ( x ), from F [ a,b ] and any scalar c , deﬁne addition and scalar multiplication as, ( f + g )( x ) = f ( x ) + g ( x ) and c f = cf ( x ), respectively. Under these operations F [ a,b ] is a vector space. Find the “zero vector” for F [ a,b ]. (3 Points )...
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