MAST 234 Notes

Mast 234: Linear Algebra & Applications

• Maple software
• Maple commands

Linear Algebra & Applications

Remember with(LinearAlgebra)

Lecture 1

review of systems of linear equations

• any row-equivalent matrix has one and only one reduced row echelon form
• A := Matrix(3,3,[1,2,3,4,5,6,7,8,0]) 3x3 Matrix
• A := RowOperation(A1, [1,2], 1/c)
• ReducedRowEchelonForm(A)
  • can use vectors together ReducedRowEchelonForm(<u|v|w|x|y|z>);to check if it is in the span
• GaussianElimination(A)
• GenerateMatrix([eq1,eq2,eq3], X)
• check the point(1,3,2) is a solution of the equations
  • subs([x1=1,x2=3,x3=2],[eq1,eq2,eq3,eq4])
• C1 = Column(A,1) or C1,C2,C3 = Column(A, 1..3)
• ReducedRowEchelonForm(<A|b>)

Lecture 2

vectors, matrices, spans

• vectors:
  • Column: X := <x1,x2,x3>, Vector([y1,y2,y3])
  • row: Vector[row]([1,c,3])
• Matrix equations as Linear combination of vectors
  • AX = b = x1C1 + x2C2 + X3C3 (Linear combination)
  • ReducedRowEchelonForm(<c1|c2|b>) get x1,x2 if has solution, then b = c1x1 + c2x2, check AX=b
  • check AX=b:
    • x1*Column(A,1)+x2*Column(A,2)+x3*Column(A,3) = b;
• 3 types of matrix
  • no solution -> inconsistent, such as 0 = 1 after use RREF (ReducedRowEchelonForm)
  • unique solution -> consistent
  • infinite solution -> consistent, such as 0 = 0 of x2 then x2 is a free variable.
  • use ReducedRowEchelonForm and GaussianElimination to check if the system is consistent or not and solutions.
• spans of vector sets
  • Span{C1,C2…Cn} = {a1C1 + a2C2 +… + anCn}
  • eg. span(u,v), u=(1,0), v=(0,1) is xu+yv=(x,y): an entire space R^2!!!
  • can use vectors together ReducedRowEchelonForm(<u|v|w|x|y|z>);to check if it is in the span
• Transpose function: Transpose(Matrix)`
  • M = <<1,2,3>|<4,5,6>> : 3X2
  • M^T = <<1,4>|<2,5>|<3,6>> : 2X3
• Matrix multiplication is use dot . not star * differently
  • A . X = v;

example:

Lecture 3

Linear dependence of vectors
Homogeneous System, Kernel, Rank

linearly dependent/independent

• linearly dependent: of there exists a non-trivial (non all zero) solution to the vector equation c1u1 + c2u2 +...+ cnun = 0
• linearly independent: only the trivial solution (c1=c2=…=cn=0)

Homogeneous system of equations:

• a system of linear equations AX=0
• A is a m X n matrix
• 0 vector is an m-dimensinal zero-vector
• any homogeneous system is always consistent, since it has at least one solution (trivial solution)
  • the column of vectors of matrix A form a linearly dependent if the linear system AX=0 has a free variable. Otherwise, the column vectors of A are linearly independent
  • Kernel (Null space) of the matrix A
• Ker(A) = {X: A.X=0} = Null(A)
• Ker(A) is never an empty set since it contains at least the zero vector 0 !!!
  • Example:
    
  • Theorems:
    
    
  • Exercise:
    
    

Linear dependence/independence of Vectors

• Exercise:
  
• example2:
  
• Example 3:
  
  • Theorems:
    

Rank

• The rank of a matrix A is the number of non-zero rows in its row echelon form, r = Rank(A)
• Example:
  • Exercise:
    
  • 如果含有parameter，要使用GaussianElimination()
    
  • 需要check，当u=0时候，在之前的题目里已经知道Rank(B)=3
    
• some important Theorems: