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Unformatted text preview: P r e l i m i n a r y d r a f t o n l y : p l e a s e c h e c k f o r fi n a l v e r s i o n ARE211, Fall 2007 LECTURE #12: TUE, OCT 9, 2007 PRINT DATE: AUGUST 21, 2007 (LINALGEBRA2) Contents 3. Linear Algebra (cont) 1 3.6. Vector Spaces 1 3.7. Spanning, Dimension, Basis 3 3.8. Matrices and Rank 7 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 satisfies 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 definition? • verbal defn excludes any straight line that doesn’t go thru the origin. 1 2 LECTURE #12: TUE, OCT 9, 2007 PRINT DATE: AUGUST 21, 2007 (LINALGEBRA2) • 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 defined 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 defined 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 ,α 2 ∈ R , α 1 x 1 + α 2 x 2 ∈ V . See how this definition 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 first 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. • 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 Berkeley.
- Fall '07