lect136_10_w09

lect136_10_w09 - Monday January 26 Lecture 10 Subspaces of...

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Monday January 26 Lecture 10 : Subspaces of R n (Refers to section 3.4) Objectives: 1. Define subspace of a vector space R n . 2. Prove that the intersection of two subspaces is a subspace. 3. Prove that the span of a finite set of vectors in R n is a subspace of R n . 10.1 Introduction – Up to this point in the course we have referred to the elements of the set R n as vectors. For reasons which we will make clear later we will now refer to R n not simply as a set but as a vector space . For example, rather than say the set R 3 we will say the vector space R 3 . 10.2 Definition – We say that a subset W of the vector space R n is a subspace of R n if W contains the vector 0 and, W is closed under linear combinations. That means, if v and w belong to W then so does α v + β w for any α and β . 10.3 Observation – The vector space R n is a subspace of itself, since it satisfies all the 2 conditions required to be a subspace. 10.4 Examples 10.4.1 Let V = R 2 . Is W = [0,1] 2 = { ( x, y ) : 0 x 1 , 0 y 1} a subspace of R 2 ? Answer: No! It is NOT closed under addition. Both (1, 1/2) and (1/2, 1) belong to W , but (1, 1/2) + (1/2, 1) = (3/2, 3/2) is not in W . 10.4.2 Example : Is W = { ( x, y, z ) in R 3 : x + y + z = 0} a subspace of R 3 ? Yes, since o W contains (0,0,0).
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o Suppose v = ( a, b, c ) and w = ( d, e, f ) are in W . Now W is all vectors x in R 3 such that < n , x > = <(1, 1, 1), x > = 0. o
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This note was uploaded on 09/04/2009 for the course MATH 136 taught by Professor All during the Winter '08 term at Waterloo.

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lect136_10_w09 - Monday January 26 Lecture 10 Subspaces of...

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