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Unformatted text preview: MA 265 GOINS REVIEW FOR MIDTERM #2 Â§ 4: Real Vector Spaces 4.1: Vectors in the Plane and 3Space â€¢ A vector in the plane or a 2vector is a 2 Ã— 1 matrix x = x y . The real numbers x and y are called the components of x . â€¢ Given two ordered pairs P ( x,y ) and Q ( x ,y ), the directed line segmentâ†’ PQ can be represented by the vector x = x x y y . We callâ†’ PQ a vector in the plane , with P the tail and Q the head . â€¢ Two vectorsâ†’ P 1 Q 1 andâ†’ 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. â€¢ 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 . â€¢ 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 . â€¢ The zero vector is the vector = . â€¢ The negative of a vector u is the vector u = ( 1) u . â€¢ The difference between two vectors u and v is the vector u v = u + ( 1) v . â€¢ A vector in space or a 3vector is a 3 Ã— 1 matrix x = ï£® ï£° x y z ï£¹ ï£» . The real numbers x , y , and z are called the components of x . â€¢ Given two ordered triples P ( x,y,z ) and Q ( x ,y ,z ), the directed line segmentâ†’ PQ can be represented by the vector x = ï£® ï£° x x y y z z ï£¹ ï£» . We callâ†’ PQ a vector in R 3 , with P the tail and Q the head . â€¢ Two vectorsâ†’ P 1 Q 1 andâ†’ P 2 Q 2 , represented by x 1 = ï£® ï£° x 1 y 1 z 1 ï£¹ ï£» and x 2 = ï£® ï£° x 2 y 2 z 2 ï£¹ ï£» , respectively, are said to be equal if their components x 1 = x 2 y 1 = y 2 , and z 1 = z 2 are equal. â€¢ The sum of two vectors u = ï£® ï£° u 1 u 2 u 3 ï£¹ ï£» and v = ï£® ï£° v 1 v 2 v 3 ï£¹ ï£» is the vector u + v = ï£® ï£° u 1 + v 1 u 2 + v 2 u 3 + v 3 ï£¹ ï£» . â€¢ The scalar multiple of a vector u = ï£® ï£° u 1 u 2 u 3 ï£¹ ï£» by a real number c is the vector c u = ï£® ï£° cu 1 cu 2 cu 3 ï£¹ ï£» . â€¢ The zero vector is the vector = ï£® ï£° ï£¹ ï£» . â€¢ The negative of a vector u is the vector u = ( 1) u . â€¢ The difference between two vectors u and v is the vector u v = u + ( 1) v . â€¢ Theorem 4.1: Let V = R 2 or R 3 . For real scalars c and d and vectors u , v , w âˆˆ V , the following properties hold: â€“ (Commutativity) u + v = v + u â€“ (Associativity) u + ( v + w ) = ( u + v ) + w and c ( d u ) = ( cd ) u â€“ (Identity) u + = + u = u and 1 u = u 1 2 MA 265 MIDTERM #2 REVIEW â€“ (Inverses) u + ( u ) = â€“ (Distributivity) c ( u + v ) = c u + c v and ( c + d ) u = c u + d u 4.2: Vector Spaces â€¢ A real vector space is a triple ( V, âŠ• , ) consisting of a set V and two operations âŠ• and with the following properties for real scalars c and d and elements u , v , w âˆˆ V : â€“ (Closure) u âŠ• v âˆˆ V and c u âˆˆ V ....
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This note was uploaded on 03/11/2010 for the course MA 261A 0026100 taught by Professor ... during the Spring '10 term at Purdue University Calumet.
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