midterm_2_topics

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

This preview shows pages 1–3. Sign up to view the full content.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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 โโโ PQ can be represented by the vector x = bracketleftbigg x โฒ โ x y โฒ โ y bracketrightbigg . We call โโโ PQ a vector in the plane , with P the tail and Q the head . โข Two vectors โโโโ P 1 Q 1 and โโโ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 โ u = ( โ 1) u . โข The difference between two vectors u and v is the vector u โ v = u + ( โ 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 โโโ PQ can be represented by the vector x = x โฒ โ x y โฒ โ y z โฒ โ z . We call โโโ PQ a vector in R 3 , with P the tail and Q the head . โข Two vectors โโโโ P 1 Q 1 and โโโ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 โ u = ( โ 1) u . โข The difference between two vectors u and v is the vector u โ v = u + ( โ 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 โ 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 + ( โ u ) = โ (Distributivity) c ( u + v ) = c u + c v and ( c + d ) u = c u + d u 4.2: Vector Spaces4....
View Full Document

{[ snackBarMessage ]}

### Page1 / 9

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online