Representation in Coordinate Systems
Explore how vectors are represented uniquely in different coordinate systems using bases, spanning vector sets, and independent vectors. Understand the significance of basis vectors and coordinate systems in vector representation and transformations.
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Presentation Transcript
Coordinate System Hung-yi Lee
Outline Coordinate Systems Each coordinate system is a viewpoint for vector representation. The same vector is represented differently in different coordinate systems. Different vectors can have the same representation in different coordinate systems. Changing Coordinates Reference: textbook Ch 4.4
Coordinate System Using Basis to represent Vector
Vector ?1=1 ?2= 1 1 1 6?1 8 4 8 4 4?2 6 2 ?1 ?2 ?2 ?1 8?1 2?2 8 4 8 4 = 8?1+ 4?2 = 6?1+ ( 2)?2
Vector 6 8 4 2 drink home ?1 ?2 ?2 ?1 ?1=1 ?2= 1 ?1=1 ?2=0 1 1 0 1
Vector ?1=1 ?2=0 2 0 ?1= ?2= 0 1 0.5 0.5 2 3 2 3 4 2.5 for left ?2 ?2 ?1 ?1 4 2?1+ 3?2= 2.5
Vector ?1=1 ?2=0 2 0 ?1= ?2= 0 1 0.5 0.5 2 3 2 3 home home 4 2.5 for left ?2 ?2 ?1 ?1 4 2?1+ 3?2= 2.5
Coordinate System A vector set B can be considered as a coordinate system for Rnif: 1. The vector set B spans the Rn Every vector should have representation 2. The vector set B is independent Unique representation B is a basis of Rn
Why Basis? Let vector set B= ?1,?2, ,?? be independent. Any vector v in Span B can be uniquely represented as a linear combination of the vectors in B. That is, there are unique scalars ?1,?2, ,??such that ? = ?1?1+ ?2?2+ + ???? Proof: ? = ?1?1+ ?2?2+ + ???? Unique? ? = ?1?1+ ?2?2+ + ???? ?1 ?1?1+ ?2 ?2?2+ + ?? ????= 0 a1 b1= a2 b2= = ak bk= 0 B is independent
Coordinate System Let vector set B= ?1,?2, ,?? be a basis for a subspace Rn B is a coordinate system For any v in Rn, there are unique scalars ?1,?2, ,??such that ? = ?1?1+ ?2?2+ + ???? B -coordinate vector of v: ?B= ( B v)
Coordinate System B= ?1,?2, ,?? vector ?B E= ?1,?2, ,?? (standard vectors) vector E is Cartesian coordinate system ( ) ? = ?E
Other System Cartesian 3 6 1 1 1 1 1 2 2 B= ?B= , , 1 1 2 7 1 1 1 1 1 2 2 = 7 5 ? = 3 + 6 2 1 1 3 6 7 8 9 1 2 3 4 5 6 C= ?C= , , 2 1 2 3 13 20 27 7 8 9 4 5 6 = ? = 3 + 6 2
Other System Cartesian Let vector set B= ?1,?2, ,?? be a basis for a subspace Rn Matrix B = ?1 ?2 ?? ?1 ?2 ?? Given ?B, how to find v? ?B= ? = ?1?1+ ?2?2+ + ???? = ? ?B (matrix-vector product)
Cartesian Other System 1 1 1 1 1 2 2 1 B= find [v]B , , 1 1 ? = 4 4 ?1 ?2 ?3 [v]B= 1 1 1 1 1 2 2 independent B is invertible (?) ? = 1 1 6 4 3 ?B= ? 1? = ? ?B= ?
Cartesian Other System Let B = {b1 , b2 , , bn} ?B= ? 1? ? ?B ?1 ?2 ?? ? = ? ?B = = c1b1 + c2b2 + + cnbn Let B= ?1,?2, ,?? is a basis of Rn. ?? B=? ?? (Standard vector)
Equation of ellipse Rotate 45 2 2 3 3 2 2 x +y = 1 ? 2 2 3 2
Equation of ellipse ?2 ?1 Use another coordinate system 2 2 2 2 B = { , } 2 2 2 2 2 3 ?1 ?2 What is the equation of the ellipse in the new coordinate system? 2 2 ( ) 2 ( ) 2 x y + = 1 3 2
Equation of ellipse 2 2 2 2 B = { , } 2 2 2 2 2 2 ( ) 2 ( ) 2 x y + = 1 3 2
Equation of hyperbola ? ? ? ? 30 ? ? 3?2+ 2?? + 3?2= 12 ?????????
Equation of hyperbola ? = ?1 ?2 1 ? 3 ? 2 3 2 2 1 2 ?1= ?2= ? 30 ? ? ?B=? ? = B ?B ? = ? ? 3 1 ? ?= ? ? 2 1 2 2 3 2 3?2+ 2?? + 3?2= 12
Equation of hyperbola ? = ?1 ?2 3?2+ 2?? + 3?2= 12 1 3 2 3 2 2 1 2 ?1= ?2= 2? 1 3 2? ? = ? =1 3 ? ? 2? + 2? ?B=? ? = ? = B ?B ? ? ? = 3 3 1 ? ?= ? ? 2 1 2 2 3 2
Summary B= ?1,?2, ,?? vector ?B E= ?1,?2, ,?? (standard vectors) vector ?B= ? 1? ? ?B ? = ? ?B
Linear Algebra Review Section 4.4 in one page Given a basis B = {b1, b2, , bn} for the vector space Rn. R2 25
Vector The same vector Represented by purple arrows Represented by red arrows
