09-basis - BASIS HOMEWORK: Section 3.2:...

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BASIS Math 21b, O. Knill HOMEWORK: Section 3.2: 6,18,24,28,48,36*,38* LINEAR SUBSPACE. A subset X of R n which is closed under addition and scalar multiplication is called a linear subspace of R n . WHICH OF THE FOLLOWING SETS ARE LINEAR SPACES? a) The kernel of a linear map. b) The image of a linear map. c) The upper half plane. d) the line x + y = 0. e) The plane x + y + z = 1. f) The unit circle. BASIS. A set of vectors ~v 1 , . . . , ~v m is a basis of a linear subspace X of R n if they are linear independent and if they span the space X . Linear independent means that there are no nontrivial linear relations a i ~v 1 + . . . + a m ~v m = 0. Spanning the space means that very vector ~v can be written as a linear combination ~v = a 1 ~v 1 + . . . + a m ~v m of basis vectors. A linear subspace is a set containing { ~ 0 } which is closed under addition and scaling. EXAMPLE 1) The vectors ~v 1 = 1 1 0 , ~v 2 = 0 1 1 , ~v 3 = 1 0 1 form a basis in the three dimensional space. If ~v = 4 3 5 , then ~v = ~v 1 + 2 ~v 2 + 3 ~v 3 and this representation is unique. We can ±nd the coe²cients by solving A~x = ~v , where A has the v i as column vectors. In our case, A = 1 0 1 1 1 0 0 1 1 x y z = 4 3 5 had the unique solution x = 1 , y = 2 , z = 3 leading to ~v = ~v 1 + 2 ~v 2 + 3 ~v 3 . EXAMPLE 2) Two nonzero vectors in the plane which are not parallel form a basis.
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This note was uploaded on 04/06/2008 for the course MATH 21B taught by Professor Judson during the Spring '03 term at Harvard.

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