Linear Algebra

1
Linear Algebra

• Basic concepts
• Matrix operations
• Gaussian elimination
• Linear independence and matrix rank

2
Basic Concepts

• m-dimensional column vector
• n-dimensional row vector
• mxn-dimensional matrix
• Square matrix m n

3
Matrix Addition Subtraction

• Only possible for matrices of same dimension
• Add/subtract matrices element-by-element
• Addition example C AB
• Subtraction example C A-B

4
Scalar and Matrix Multiplication

• Scalar multiplication
• B kA
• Dimensions
• General formula
• Example
• Matrix multiplication
• C AB
• Only possible if the number of columns of A is
  equal to the number of rows of B

5
Matrix Multiplication cont.

• General representation
• Dimensions
• Formula
• Examples
• Noncommutative operation

6
Transpose

• Notation B AT
• Dimensions
• Formula
• Example
• Important properties

7
Common Matrices

• Symmetric matrix AT A
• Skew-symmetric matrix AT -A
• Example of a diagonal matrix
• Examples of triangular matrices
• Identity matrix

8
Systems of Linear Algebraic Equations

• Scalar representation
• Matrix representation Ax b
• Homogeneous system b 0
• One obvious solution x 0

9
Triangular Systems

• Example
• Solution
• Gaussian elimination
• Transform original system into diagonal form
• Accomplished by elementary row operations

10
Gaussian Elimination

• Augmented matrix
• Elementary row operations
• Interchange of two rows
• Multiplication of a row by a non-zero constant
• Addition of a constant multiple of one row to
  another row
• Operations on columns are not allowed!

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Gaussian Elimination Example

• Form augmented matrix
• Eliminate x1 from second and third equations

12
Gaussian Elimination Example

• Eliminate x2 from third equation
• Solve triangular system

13
Three Possible Cases

• Uniquely determined system
• Same number of equations unknowns
• No degrees of freedom
• Usually yields a unique solution
• Underdetermined system
• More unknowns than equations
• Extra degrees of freedom
• Yields infinite number of solutions
• Overdetermined system
• More equations than unknowns
• Usually yields no solution (inconsistent)

14
Linear Independence

• Given m vectors a(1), a(2), , a(m) of equal
  dimension
• Consider the linear equation
• Linear independent vectors
• Equation satisfied only for cj 0
• Each vector is unique
• Linear dependent vectors
• Equation also satisfied for some non-zero cj
• At least one vector is redundant
• Example of linearly dependent vectors

15
Matrix Rank

• r rank(A)
• Number of linearly independent row vectors of A
• Number of linearly independent column vectors of
  A
• Examples
• Rank can be determined through elementary row
  operations (see text)