Chapter 2. Vector Space (2-4 ~ 2-6)

Transcription

1 Chapter 2. Vector Space (2-4 ~ 2-6) KAIS wit Lab 송유재

2 he four fundamental subspaces Column space C( A) : all linear combination of column vector Dimension : rank r Nullspace ( ): all vectors x such that Dimension : n r N A Ax 0 Row space C( A ) : the column space of Dimension : r A A 1 0 0, r Left nullspace N( A ) : the nullspace of Dimension : m r A 2

3 Subspaces for A and U Our problem will be connect the four space for U (after elimination) to the four spaces for A U A the row space of A A and U have different rows, but the combitation of the rows are identical : same space 3

4 Subspaces for A and U U A the nullspace of A the nullspace of A is the same as nullspace of U and R nullspace is kernel, n r dimension is nullity 3 the column of A hose spaces are different, but their dimensions are same the pivot column of A are a basis for its column space 4

5 Subspaces for A and U U A the left nullspace of A dimension of C( A ) + dimension of N( A) = number of columns 5

6 Existence of inverse Existence : full row rank r=m. Ax b has at least one solution x for m every b if and only if the columns span. hen A has a rightinverse C such that. his is possible only if m n AC I m Uniqueness : full column rank r=n. Ax b has at most one solution x for every b if and only if the columns are linearly independent. hen A has an left inverse B such that BA I n Best left and right inverse B( A A) A, C A ( AA ) 1 1 Ex) A

7 Matrices of rank 1 Every matrix of rank 1 has the simple form A uv = column times row. A

8 Graph GNE (, ) Graph and incidence matrix consists of a set of vertices or nodes, and a set of edges that connect them If edge has the direction of the arrow, it is directed Incidence matrix shows the relationship between two classes of object A

9 Four subspace for incidence matrix Nullspace of A he nullspace contains x , since Ax 0 he equation Ax bdoes not have a unique solution. Any constant vector x c c c c can be added to any particular solution of Ax b his has a meaning if we think x1, x2, x3, x4 as the potentials at the node he five components of Ax give the differences in potential across the five edge 9

10 Four subspace for incidence matrix Column space of A For which differences b, b, b, b, b can we solve Ax b? b x b 2 x b 3 x b4 x b 5 If b is in the column space, the b x b2 ( b1b3) x b 3 x b4 ( b3 b5) x b b b b 0 and b b b 0 he test for b to be in the column space is Kirchhoff s voltage law : he sum of potential differences around a loop must be zero 10

11 Four subspace for incidence matrix Left nullpsace of A he vector y has five components hese numbers represent currents flowing along the five edges Ay 0 y y 0 otalcurrent to node1is zero 1 2 y y y 0 to node y y y 0 to node3 y y to node4 he simplest solution are currents around small loops : y , y

12 Four subspace for incidence matrix Row space of A he vector f has four components he numbers represent current source into nodes y y1 y2 f1 y 2 Ayf y1y3y 4 f 2 y y2y3y 5 f 3 y y4y5 f4 y 5 he source must balance y y f1 1 2 he equations Ayf at nodes express Kirchhoff s current law: the net current into every node is zero. Flow in = Flow out 12

13 Networks Network A graph becomes a network when number c,..., 1 cm are designed to the edges c i : the length of edge i, or conductance hose number go into diagonal matrix C Reflects material properties Ohm s Law : e i y : voltage drop Ax i ce y Ce i i gives the drop in potential Part of that drop may be due to a battery in the edge of strength b i 1 y C( bax) or C yaxb 13

14 Networks he fundamental equations of equilibrium Combine Ohm and Kirchhoff into a central problem of applied mathematics Equilibrium equations Block form 1 C Ay b A 0 x f 1 C y Ax b Ay f y b 0 x ACA f A Cb 1 C A Fundamental equation A CAx A Cb f 14

15 Networks Important remark One potential must be fixed in advance x: n 0 he nth node is grounded Resulting matrix is what we now mean A: its n 1 column are independent Example 1. 15

16 he meaning of transformation Linear transformation When A multiplies x, it transforms that vector into a new vetor. Example 16

17 Linear transformation ransformation that obey the rule of linearity are called linear transformation he example of linear transformation Differentiation, integration 17

18 ransformation represented by matrix Finding matrices that represent differentiation In case of polynomials of degree 3 ex) p 2 tt t

19 ransformation represented by matrix Finding matrices that represent integration In case of polynomials of degree 3 ex) p 2 tt t

20 Rotations Q, Projections P, Reflections H Rotation through Projection onto line 20

21 Rotations Q, Projections P, Reflections H Reflection through the line 21