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

Basic Concepts m-dimensional column vector n-dimensional row vector mxn-dimensional matrix Square matrix: m = n

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

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

Matrix Multiplication cont. General representation Dimensions: Formula Examples Noncommutative operation:

Transpose Notation: B = AT Dimensions: Formula: Example Useful properties

Common Matrices Symmetric matrix: AT = A Skew-symmetric matrix: AT = -A Example of a diagonal matrix Examples of triangular matrices Identity matrix

Matlab Examples Matrix addition >> A=[1 3 2; 2 4 5]; >> D=A+B D = 4 -1 8 3 2 10 Matrix multiplication >> C=[2 3; -1 2; 4 -3]; >> E=A*C E = 7 3 20 -1

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

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

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!

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

Gaussian Elimination Example Eliminate x2 from third equation Solve triangular system

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)

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

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)

Matlab Example Ax = b  x = A-1b (discuss next lecture) >> A=[3 -2 2; -5 4 -3; -4 3 -2]; >> rank(A) ans = 3 >> b=[-1; 3; 1]; >> x=inv(A)*b x = 3.0000 3.0000 -2.0000