Linear Systems - Advanced Engineering Math - Tutorial Slides, Slides of Engineering Mathematics

Download Linear Systems - Advanced Engineering Math - Tutorial Slides and more Slides Engineering Mathematics in PDF only on Docsity!

Linear Systems, linear dependence

• Brief review
  • Solving linear system: Gaussian Elimination, Reduced Row

Echelon

• Dot product, Matrix-Vector Production, Matrix-Matrix

Productions

• Linear Dependence, number of solutions of linear system.
• Drill problems and solutions
• Q&A

Row Echelon Form (REF)

• A rectangular matrix is in row echelon form if
  • All nonzero rows are above any all-zero row.
  • All entries below a leading entry are zeros.
  • In any pair of adjacent nonzero row, say row i and row i+1, the leading entry in row i is to the left of the leading entry of row i+1.

• Dot Production
  • It measures the “angle” between two vectors
  • Property:

a c (^) ac bd a (^2) b (^2) c (^2) d (^2) cos b d

θ       ⋅^  =^ +^ =^ +^ ⋅^ +^ ⋅    

• Matrix-Vector Production
• Matrix-Matrix Productions

Linear Combination

• Given vectors v 1 , v 2 , …, v i in , and i real number c 1 , c 2 , …, c (^) i, the vector w obtained by w = c 1 v 1 + c 2 v 2 + …+ c (^) i v i is called a linear combination of v 1 , v 2 , …, v i.
• Question 1: determine if β is a linear combination of^ α 1 ,^ α^2 ,α 3

1 2 3

2 1 1 2 3 , 1 , 2 , 3 1 2 3 5

β α α α

     −    = ^ ^ = ^ − ^ = ^ ^ = ^ −           (^) −     (^) −   

Linear dependence

• v 1 , v 2 , …, v r are said to be linear dependent if we can find r real number c 1 , c 2 , …, c (^) r, not all of them equal to zero, such that 0 = c 1 v 1 + c 2 v 2 + …+ c (^) r v r

Otherwise, v 1 , v 2 , …, v r are said to be linear independent.

• In other words, v 1 , v 2 , …, v r are be linear independent if, the only choice of c 1 , c 2 , …, c (^) r, such that 0 = c 1 v 1 + c 2 v 2 + …+ c (^) r v r is c 1 = c 2 = …= c (^) r=0.

Number of solutions of linear system

At most one solution (^) At least one solution

Ax = b m equations n variables

Every vector in is a linear combination of the columns in A.

Columns of A are linearly independent

The columns of A contain a lot of information about the nature of the solutions.

Solution:

If the set of solution is nonempty, then so or

Solution:

When , the solution is

When , the solution is

Solution:

Assume For is linear dependent, there exist

with x 1^ , x 2 not both 0, and

then:

For we have

Solution:

As for should not be both 0, so

( ) ( ) ( )( )

1 1 1 2 1 2 (^1 ) 2 1 2 2 2 2 1 2 1 1 2

+1 0 +1 (^0) +1 0 +1 0 +1 +1 +1 (^00) + +1 0

k k k k k k k k k k k k k^ k^ k^ k k k

 ^ ^ ^ ^    →^ ^ ^ →^    (^)    

then:

or: