# lecture09 - Lecture Lecture 9 3.1 Euclidean n-spaces 3.2...

This preview shows pages 1–12. Sign up to view the full content.

ecture 9 Lecture 9 3.1 Euclidean n-spaces 3.2 Linear Combinations and Linear Spans Chapter 3 Vector Spaces 1

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

View Full Document
ap of LA an n ma tr ix Map of LA A is an n ä n matrix A is invertible A is not invertible rref of A is identity matrix rref of A has a zero row det A 0 det A = 0 Ax = 0 has only the trivial solution Ax = 0 has non-trivial solutions Ax = b has a unique solution Ax = b has no solution or infinitely many solutions to be continued Lecture 9 Maps of LA 2
ection 3 1 Section 3.1 Euclidean n-Spaces Objectives hat is an - ector What is an n vector ? • What are some operations on n-vectors? •Whatisa Euclidean n-space R n ? ow to express subsets of • How to express subsets of R n ? Chapter 3 Vector Spaces 3

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

View Full Document
What is a vector? Discussion 3.1.1 - 3.1.2 (Vectors) Notation geometric algebraic (2-dimension) (3-dimension) v u u w (v 1 , v 2 ) (v 1 , v 2 , v 3 ) Vector u (u 1 , u 2 ) (u 1 , u 2 , u 3 ) u v u v u + v Addition u + v u v (u 1 +v 1 , u 2 +v 2 ) (u 1 +v 1 , u 2 +v 2 , u 3 +v 3 ) (-u , -u ) u u u 2 1 u 2 Negative - u calar 1 2 (-u 1 , -u 2 , -u 3 ) (au , au ) Chapter 3 Vector spaces 4 u u ) . ( 5 1 Scalar multiple a u ( 1 , 2 ) (au 1 , au 2 , au 3 )
Geometric vs algebraic vectors Discussion 3.1.2.1 Position a geometric vector u in the xy -plane such that its initial point is at the origin (0, 0). y u ( u 1 , u 2 ) algebraic vector 0) In the xy-plane, ( u 1 , u 2 ) can represent oth a oint nd an rrow x (0, 0) Chapter 3 Vector spaces 5 both a point and an arrow . Similarly for the 3-space.

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

View Full Document
(u 1 , u 2 ) (u , u , u ) 2-vector - ector What is an n-vector? Definition 3.1.3 ( 1 , 2 , 3 ) 3 vector ( u 1 , u 2 , …, u i , , u n ) where u 1 , u 2 , …, u n are real numbers n -vector or ordered n -tuple of real numbers i th component (or i th coordinate ) Chapter 3 Vector spaces 6
Four dimension Spacetime ) ( t , z , y , x 4-vectors lbert Einstein Chapter 3 Vector spaces 7 Albert Einstein Special Relativity Theory

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

View Full Document
Four dimensional cube ne imensional ne segment a one-dimensional line segment a two-dimensional square a three-dimensional cube ur- imensional ube Chapter 3 Vector spaces 8 a four dimensional cube (tesseract)
n -vectors as matrices Notation 3.1.5 e can identify an - ector ( with We can identify an n vector u 1 , u 2 , …, u n ) a 1 x n matrix ( u 1 u 2 u n ) ( row vectors ) u u 2 1 or an n x 1 matrix ( column vector ). n u # arning: o not use two different sets of Warning: Do not use two different sets of notation for n -vectors within the same context. Chapter 3 Vector spaces 9 Definitions and properties of vector operations are similar to matrix operations (see 3.1.3 to 3.1.6 )

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

View Full Document
Lost in What is a Euclidean n-space ? Definition 3.1.7 The set of all n -vectors of real numbers space is called the Euclidean n -space and is denoted by R n . u œ R n u = ( u 1 , u 2 , …, u n ) u is an n-vector Euclidean 2-space XY-plane uclidean 3- pace YZ- pace Euclidean 3 space XYZ space Chapter 3 Vector spaces 10
How to express subsets of R n ? Set notation Example 3.1.8.1 S = { ( u 1 , u 2 , u 3 ) | u 1 = 0 and u 2 = u 3 } implicit form ( u , u , u ) is an element of the subset S subset of R 3 conditions on the components , 0, 0), (0, 1, 1), (0, 1 , 2 , 3 ) if and only if u 1 = 0 and u 2 = u 3 .

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.

{[ snackBarMessage ]}

### Page1 / 32

lecture09 - Lecture Lecture 9 3.1 Euclidean n-spaces 3.2...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online