Linear Algebra 1.Basic concepts 2.Matrix operations.

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Presentation transcript:

Linear Algebra 1.Basic concepts 2.Matrix operations

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

Matrix Addition & Subtraction l Only possible for matrices of same dimension l Add/subtract matrices element-by-element l Addition example: C = A+B l Subtraction example: C = A-B

Scalar and Matrix Multiplication l Scalar multiplication »B = kA »Dimensions: »General formula: »Example l 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. l General representation »Dimensions: »Formula l Examples l Noncommutative operation:

Transpose l Notation: B = A T l Dimensions: l Formula: l Example l Important properties

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

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

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

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