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Arlon
Online Engineering
Library - TechRef!
Linear Algebra
MATRICES
A matrix is an ordered rectangular array of numbers with m rows and n columns. The element aij refers to row i and column j.
Multiplication
If A = (aik) is an m x n matrix and B = (bkj) is an n x s matrix, the matrix product AB is an m x s matrix
where n is the common integer representing the number of columns of A and the number of rows of B (l and k = 1,2,...,n).
Addition
If A = (aij) and B = (bij) are two matrices of the same size m x n, the sum A + B is the m x n matrix C = (cij) where cij = aij + bij.
Identity
The matrix I = (aij) is a square n x n identity matrix where aii =1 for i = 1,2,...,n and aij = 0 for i != j.
Transpose
The matrix B is the transpose of the matrix A if each entry bji in B is the same as the entry aij in A and conversely. In equation form, the transpose is B = AT.
Inverse
The inverse B of a square n x n matrix A is
adj(A) = adjoint of A (obtained by replacing AT elements with their cofactors, see Determinants) and |A| = determinant of A.
Determinant
A determinant of order n consists of n2 numbers, called the elements of the determinant, arranged in n rows and n columns and enclosed by two vertical lines. In any determinant, the minor of a given element is the determinant that remains after all the elements are struck out that lie in the same row and in the same column as teh given element. Consider an element which lies in the hth column and the kth row. The cofactor of this element is the value of the minor of the element (if h + k is even), and it is the negative of the value of the minor of the element (if h + k is odd).
If n is greater than 1, the value of a determinant of order n is the sum of the n products formed by multiplying each element of some specified row (or column) by its cofactor. This sum is called the expansion of the determinant [according to the elements of the specified row (or column)]. For a second-order determinant:
For a third-order determinant:
VECTORS
A vector is a one-dimensional array, usually associated with quantities defined by a magnitude and direction.
a = axi + ayj + azk
Addition and Subtraction
a + b = (ax + bx)i + (ay + by)j + (az + bz)k
a + b = (ax - bx)i + (ay - by)j + (az - bz)k
Dot Product
The dot product is a scalar product and represents the projection of b onto a times ||a||. It is given by
a · b = axbx + ayby + azbz
= ||a|| ||b|| cos  = b · a
Cross Product
The cross product is a vector product of magnitude ||b|| ||a|| sin which is perpendicular to the plane containing a and b. The product is
The sense of a x b is determined by the right-hand rule.
a x b = ||a|| ||b|| n sin , where
n = unit vector perpendicular to the plane of a and b.
Identities
a · b = b · a; a · (b + c) = a · b + a · c
a · a = ||a||2
i · i = j · j = k · k = 1
i · j = j · k = k · i = 0
if a · b = 0, then either a = 0, b = 0, or a is perpendicular to b.
a x b = -b x a
a x (b + c) = (a x b) + (a x c)
(b + c) x a = (b x a) + (c x a)
i x i = j x j = k x k = 0
i x j = k = -j x i; j x k = i = -k x j; k x i = j = -i x k
if a x b = 0, then either a = 0, b = 0, or a is parallel to b.
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