mathLinAlgebra2-07 - ARE211, Fall 2007 LINALGEBRA2: TUE,...

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ARE211, Fall 2007 LINALGEBRA2: TUE, OCT 9, 2007 PRINTED: OCTOBER 18, 2007 (LEC# 12) Contents 3. Linear Algebra (cont) 1 3.6. Vector Spaces 1 3.7. Spanning, Dimension, Basis 4 3.8. Matrices and Rank 8 3. Linear Algebra (cont) 3.6. Vector Spaces A key concept in linear algebra is called a vector space. What is it? It is a special kind of set of vectors, satisfying a particular property. The property is a bit abstract, so we’ll work up to it: Verbal Defn: a nonempty set of vectors V is called a vector space if it satisFes the following property: given any two vectors that belong to the set V , every linear combination of these vectors is also in the space. One kind of vector space is a set of vectors represented by a line. Note that a line can (and should) be thought of as just another set of vectors. Think about all of the lines you can draw in R 2 . Which lines are vector spaces, according to the above deFnition? 1
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2 LINALGEBRA2: TUE, OCT 9, 2007 PRINTED: OCTOBER 18, 2007 (LEC# 12) verbal defn excludes any straight line that doesn’t go thru the origin. verbal defn excludes any line that “curve.” verbal defn excludes any straight line thru the origin that “stops.” What’s left is rays thru the origin. Every ray through the origin is a vector space. Mathematically, any ray through the origin is deFned as the set of all scalar multiples of a given vector. It’s called a “one-dimensional vector space”, because it is “constructed” from a single vector. Note that neither the nonnegative or the positive cones deFned by a single vector are vector spaces. Turns out that there’s exactly two more vector spaces in R 2 : The set consisting of the entire plane is a vector space, i.e., R 2 . This is a two dimensional vector space. The set consisting of zero alone is a vector space. This is a zero dimensional vector space. Math Defn: a nonempty set of vectors V is called a vector space if for all x 1 ,x 2 V , for all α 1 2 R , α 1 x 1 + α 2 x 2 V . See how this deFnition stacks up against our examples: look at a line that curves: take any vector in the set consisting of the points in the line. Let v 1 , v 2 be any vectors in the line. Take zero times the Frst and 2 times the second. Is it in the set? No. Conclude that the curved line is not a vector space. look at a straight line that doesn’t pass through the origin. Let v 1 , v 2 be any vectors in the line. Take zero times both. In the set? No. Conclude that a straight line that doesn’t pass through the origin is not a vector space.
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ARE211, Fall 2007 3 take any straight line thru the origin that “stops.” Let v 1 , v 2 be any vectors in the line.
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This note was uploaded on 08/01/2008 for the course ARE 211 taught by Professor Simon during the Fall '07 term at University of California, Berkeley.

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mathLinAlgebra2-07 - ARE211, Fall 2007 LINALGEBRA2: TUE,...

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