MATH 310
Self-quiz 3
1. Determine whether the following subset Uof R3is a linear subspace or
not:
U={(x, y, z)T:x−2y= 0z= 3y}
2. Examine if the vector (1,0,−3)Tis a linear combination of the vectors
(1,1,0)Tand (2,−3,0)T.
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Questions with Solution - Applied Linear Algebra | Quiz 3 | MATH 310, Quizzes of Linear Algebra
Material Type: Quiz; Class: Applied Linear Algebra; Subject: Mathematics; University: University of Illinois - Chicago; Term: Unknown 2012;
Typology: Quizzes
2011/2012
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MATH 310
Self-quiz 3
- Determine whether the following subset U of R^3 is a linear subspace or not:
U = {(x, y, z)T^ : x − 2 y = 0z = 3y}
- Examine if the vector (1, 0 , −3)T^ is a linear combination of the vectors (1, 1 , 0)T^ and (2, − 3 , 0)T^.
MATH 310
Self-quiz 3
- Determine whether the following subset U of R^3 is a linear subspace or not:
U = {(x, y, z)T^ : x − 2 y = 0z = 3y}
Solution: We need to examine if the sum of any two vectors of U also belongs to U and also if any scalar multiple of any vector of U also belongs to U.
Suppose that v, w belong to U. To fix ideas let:
v = (x 1 , y 1 , z 1 )T^ w = (x 2 , y 2 , z 2 )T
Then v + w = (x 1 + x 2 , y 1 + y 2 , z 1 + z 2 )T^. We need to test whether:
(x 1 + x 2 ) − 2(y 1 + y 2 ) = 0 z 1 + z 2 = 3(y 1 + y 2 )
Since v is in U , we have that x 1 − 2 y 1 = 0. Since w belongs to U , we also have that x 2 − 2 y 2 = 0. If we add these two relations, then we get:
x 1 + x 2 − 2 y 1 − 2 y 2 = 0
which is really the first relation that needs to be satisfied.
Also, we know that z 1 = 3y 1 and z 2 = 3y 2. This is because v and w belong to U. If we add these two relations, then we get:
z 1 + z 2 = 3y 1 + 3y 2
which is really the second sought-for relation.