Vectors. Vector Space  A vector space is an abstract mathematical object.  It has a set of vectors. Commutative with additionCommutative with addition.

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Presentation transcript:

Vectors

Vector Space  A vector space is an abstract mathematical object.  It has a set of vectors. Commutative with additionCommutative with addition Matrices, polynomials, functionsMatrices, polynomials, functions  It has a set of scalars. Objects with addition and multiplicationObjects with addition and multiplication Real numbers, complex numbersReal numbers, complex numbers  Vector spaces all have a property of scalar multiplication. Vectors v, u Scalars f, g  Scalar multiplication has special features. fv is a vector and 1v = v f(gv) = (fg)v f(v+u) = fv + fu (f+g)v = fv + gv

Linear Operator  A linear operator transforms a vector. Operator AOperator A Vectors v, uVectors v, u Scalar fScalar f  The linear properties of addition and scalar multiplication are preserved. A(fv) = f (Av)A(fv) = f (Av) A(v+u) = Av + AuA(v+u) = Av + Au

Cartesian Vector  A 3-D Cartesian vector is an ordered set of real numbers. ( x 1, x 2, x 3 ) Indices indicate components Addition by component Scalar multiplication on each component  This specific type of a vector is what we think of as having a “magnitude and direction”. x1x1 x3x3 x2x2

Summation Rule  The index for a vector component can be written as a variable.  Any index written twice within a multiplicative expression implies summation. Transformations with multiple indicesTransformations with multiple indices Kronecker deltaKronecker delta

Coordinate Transformation  A vector can be described by many Cartesian coordinate systems. Transform from one system to another Transformation matrix L x1x1 x2x2 x3x3 A physical property that transforms like this is a Cartesian vector.

Inner Product  The inner product of two vectors is a scalar. Summation over index Dot product  Components are the result of projection. Inner product with bases Bases product as delta  The square of the magnitude is its inner product with itself. x3x3

Vector Product  The vector product applies to any two vectors. Cross productCross product Wedge productWedge product  The vector product is not commutative. Result perpendicular to planeResult perpendicular to plane Reversing order is anti- parallelReversing order is anti- parallel

Permutation  The vector product can be defined by components. Permutation epsilon:  ijk = 0, any i, j, k the same  ijk = 1, if i, j, k an even permutation of 1, 2, 3  ijk = -1, if i, j, k an odd permutation of 1, 2, 3 Add clockwise Subtract counterclockwise next