midterm_2_topics - MA 265 GOINS REVIEW FOR MIDTERM#2 ย 4...

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Unformatted text preview: MA 265 GOINS REVIEW FOR MIDTERM #2 ยง 4: Real Vector Spaces 4.1: Vectors in the Plane and 3-Space โ€ข A vector in the plane or a 2-vector is a 2 ร— 1 matrix x = bracketleftbigg x y bracketrightbigg . The real numbers x and y are called the components of x . โ€ข Given two ordered pairs P ( x,y ) and Q ( x โ€ฒ ,y โ€ฒ ), the directed line segment โˆ’โˆ’โ†’ PQ can be represented by the vector x = bracketleftbigg x โ€ฒ โˆ’ x y โ€ฒ โˆ’ y bracketrightbigg . We call โˆ’โˆ’โ†’ PQ a vector in the plane , with P the tail and Q the head . โ€ข Two vectors โˆ’โˆ’โˆ’โ†’ P 1 Q 1 and โˆ’โˆ’โˆ’โ†’ P 2 Q 2 , represented by x 1 = bracketleftbigg x 1 y 1 bracketrightbigg and x 2 = bracketleftbigg x 2 y 2 bracketrightbigg , respectively, are said to be equal if their components x 1 = x 2 and y 1 = y 2 are equal. โ€ข The sum of two vectors u = bracketleftbigg u 1 u 2 bracketrightbigg and v = bracketleftbigg v 1 v 2 bracketrightbigg is the vector u + v = bracketleftbigg u 1 + v 1 u 2 + v 2 bracketrightbigg . โ€ข The scalar multiple of a vector u = bracketleftbigg u 1 u 2 bracketrightbigg by a real number c is the vector c u = bracketleftbigg cu 1 cu 2 bracketrightbigg . โ€ข The zero vector is the vector = bracketleftbigg bracketrightbigg . โ€ข The negative of a vector u is the vector โˆ’ u = ( โˆ’ 1) u . โ€ข The difference between two vectors u and v is the vector u โˆ’ v = u + ( โˆ’ 1) v . โ€ข A vector in space or a 3-vector is a 3 ร— 1 matrix x = x y z . The real numbers x , y , and z are called the components of x . โ€ข Given two ordered triples P ( x,y,z ) and Q ( x โ€ฒ ,y โ€ฒ ,z โ€ฒ ), the directed line segment โˆ’โˆ’โ†’ PQ can be represented by the vector x = x โ€ฒ โˆ’ x y โ€ฒ โˆ’ y z โ€ฒ โˆ’ z . We call โˆ’โˆ’โ†’ PQ a vector in R 3 , with P the tail and Q the head . โ€ข Two vectors โˆ’โˆ’โˆ’โ†’ P 1 Q 1 and โˆ’โˆ’โˆ’โ†’ 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. โ€ข 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 . โ€ข 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 . โ€ข The zero vector is the vector = . โ€ข The negative of a vector u is the vector โˆ’ u = ( โˆ’ 1) u . โ€ข The difference between two vectors u and v is the vector u โˆ’ v = u + ( โˆ’ 1) v . โ€ข Theorem 4.1: Let V = R 2 or R 3 . For real scalars c and d and vectors u , v , w โˆˆ V , the following properties hold: โ€“ (Commutativity) u + v = v + u โ€“ (Associativity) u + ( v + w ) = ( u + v ) + w and c ( d u ) = ( cd ) u โ€“ (Identity) u + = + u = u and 1 u = u 1 2 MA 265 MIDTERM #2 REVIEW โ€“ (Inverses) u + ( โˆ’ u ) = โ€“ (Distributivity) c ( u + v ) = c u + c v and ( c + d ) u = c u + d u 4.2: Vector Spaces4....
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midterm_2_topics - MA 265 GOINS REVIEW FOR MIDTERM#2 ย 4...

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