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

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ecture 9 Lecture 9 3.1 Euclidean n-spaces 3.2 Linear Combinations and Linear Spans Chapter 3 Vector Spaces 1
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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
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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
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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 )
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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.
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(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
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Four dimension Spacetime ) ( t , z , y , x 4-vectors lbert Einstein Chapter 3 Vector spaces 7 Albert Einstein Special Relativity Theory
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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)
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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 )
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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
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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 .
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lecture09 - Lecture Lecture 9 3.1 Euclidean n-spaces 3.2...

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