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Unformatted text preview: MA 265 GOINS REVIEW FOR MIDTERM #2 4: Real Vector Spaces 4.1: Vectors in the Plane and 3Space A vector in the plane or a 2vector is a 2 1 matrix x = x y . The real numbers x and y are called the components of x . Given two ordered pairs P ( x,y ) and Q ( x ,y ), the directed line segment PQ can be represented by the vector x = x x y y . We call PQ a vector in the plane , with P the tail and Q the head . Two vectors P 1 Q 1 and P 2 Q 2 , represented by x 1 = x 1 y 1 and x 2 = x 2 y 2 , respectively, are said to be equal if their components x 1 = x 2 and y 1 = y 2 are equal. The sum of two vectors u = u 1 u 2 and v = v 1 v 2 is the vector u + v = u 1 + v 1 u 2 + v 2 . The scalar multiple of a vector u = u 1 u 2 by a real number c is the vector c u = cu 1 cu 2 . The zero vector is the vector = . The negative of a vector u is the vector u = ( 1) u . The difference between two vectors u and v is the vector u v = u + ( 1) v . A vector in space or a 3vector is a 3 1 matrix x = x y z . The real numbers x , y , and z are called the components of x . Given two ordered triples P ( x,y,z ) and Q ( x ,y ,z ), the directed line segment PQ can be represented by the vector x = x x y y z z . We call PQ a vector in R 3 , with P the tail and Q the head . Two vectors P 1 Q 1 and P 2 Q 2 , represented by x 1 = x 1 y 1 z 1 and x 2 = x 2 y 2 z 2 , respectively, are said to be equal if their components x 1 = x 2 y 1 = y 2 , and z 1 = z 2 are equal. The sum of two vectors u = u 1 u 2 u 3 and v = v 1 v 2 v 3 is the vector u + v = u 1 + v 1 u 2 + v 2 u 3 + v 3 . The scalar multiple of a vector u = u 1 u 2 u 3 by a real number c is the vector c u = cu 1 cu 2 cu 3 . The zero vector is the vector = . The negative of a vector u is the vector u = ( 1) u . The difference between two vectors u and v is the vector u v = u + ( 1) v . Theorem 4.1: Let V = R 2 or R 3 . For real scalars c and d and vectors u , v , w V , the following properties hold: (Commutativity) u + v = v + u (Associativity) u + ( v + w ) = ( u + v ) + w and c ( d u ) = ( cd ) u (Identity) u + = + u = u and 1 u = u 1 2 MA 265 MIDTERM #2 REVIEW (Inverses) u + ( u ) = (Distributivity) c ( u + v ) = c u + c v and ( c + d ) u = c u + d u 4.2: Vector Spaces A real vector space is a triple ( V, , ) consisting of a set V and two operations and with the following properties for real scalars c and d and elements u , v , w V : (Closure) u v V and c u V ....
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 Spring '10
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