1. Linear independence and matrix rank
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 defined for matrices of same dimension
Add/subtract matrices element-by-element
Addition example: C = A+B
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
Useful 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. Matlab Examples Matrix addition >> A=[1 3 2; 2 4 5];
>> D=A+B
D =
Matrix multiplication
>> C=[2 3; -1 2; 4 -3];
>> E=A*C
E =

9. Systems of Linear Algebraic Equations
Scalar representation
Matrix representation: Ax = b
Homogeneous system: b = 0
One obvious solution: x = 0

10. Triangular Systems Example Solution Gaussian elimination
Transform original system into diagonal form
Accomplished by elementary row operations

11. 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!

12. Gaussian Elimination Example
Form augmented matrix
Eliminate x1 from second and third equations

13. Gaussian Elimination Example
Eliminate x2 from third equation
Solve triangular system

14. Three Possible Cases Uniquely determined system Underdetermined system
Same number of equations and 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)

15. 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

16. Matrix Rank r = rank(A) Examples
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)
Square matrix must be full rank for linear algebraic system to yield unique solution (next lecture)

17. Matlab Example
Ax = b  x = A-1b (discuss next lecture) >> A=[3 -2 2; ; ]; >> rank(A) ans = 3 >> b=[-1; 3; 1]; >> x=inv(A)*b x =