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Algebra means the reunion of broken parts. It began with computations similar to arithmetic, but numbers were replaced by letters. For example if we say aX+bY+c=0, we have the equation of a line. You may put any number for a, b, and c and it holds true forever as a line. For what is required to know of algebra in the FE exam, it starts with the Logarithms. Simply, it can be shown as the following:
The common logs have a base of 10 (not written) while the base for natural logs is the Euler's number (e ≈ 2.71828) and it is written as Ln. Sometimes, we need to simplify expressions that contain logarithmic and exponential equations, for which, logarithmic identities (accessed 09/15/15)are good to know.
The next chapter in algebra is about the complex numbers which can be found in this page (accessed 09/15/15).
next is the subject of matrices. A matrix with n rows and m columns is shown as a m x n matrix. For a square matrix, m = n and is known as the order of matrix. In scalar multiplication of matrix, all entries of the matrix is multiplied by that scaler resulting in the result matrix. Two matrices can be multiplied to each other if the number of columns in first matrix is equal to the number of rows of the second matrix. In addition of two matrices, the two matrices must have the same shape and size so each entry in the first matrix can be added to the corresponding entry in the second matrix.
A diagonal matrix is a square matrix that has all entries as zero except the aij entries where i = j. A triangular matrix is when all entries below or above the diagonal are zeros. The identity matrix, I, is a diagonal matrix that the values on its diagonal is all 1, and for a square matrix A we have AI = IA = A. Transpose of matrix A (n x m) is shown as AT (m x n).
Determinant of matrix A is only defined for square matrices which is a scalar shown as D{A}, Det{A}, or |A|. There are several properties for matrices related to their determinant (accessed 09/15/15) that can be see in this page (accessed 09/15/15). In a square matrix, minor of a particular entry is the submatrix remains in the original when the associated row and column of the entry aij is removed from the original matrix. Cofactors are the minors associated with particular entries multiplied by +1 (if i+j is even) or -1 (if i + j is odd). Here is a nice video (accessed 09/15/15) that explains the cofactors and minors with examples. If determinant of a matrix is zero, the matrix is called singular. Transpose of the cofactor matrix is known as classical adjoint Aadj, Adj{A} matrix.
Inverse of matrix A is shown as A-1 and defined for only non-singular square matrices in a way that the product of matrix A by its inverse results in identity matrix. This video (accessed 09/15/15)shows how to calculate the inverse of a 2 x 2 matrix and this video (accessed 09/15/15) shows how to calculate the inverse of a 3 x 3 matrix
Matrices are also used in solving simultaneous linear equations, where they constitute the coefficient, variables, and constant matrices. In such cases, determinants of the matrices can be used to solve the system of equations by using the Cramer's rule (accessed 09/15/15).
Next is the concept of vector. A physical property can be shown as a scalar (mass, enthalpy, speed, density, etc.), vector (displacement, momentum, force, etc.), or a tensor. Scalars only have magnitude while vectors have magnitude and direction. Two vectors with the same magnitude and direction are equal even their lines of action or points of application (terminal point) are not the same. Unit vectors have unit magnitude (of 1) with standard cartesian unit vectors of i, j, and k in the x, y, and z directions of cartesian coordinate system. Vector a can be written in terms of its components and unit vectors like A = axi + ayj + azk. A tensor has a magnitude and a direction but the direction is not unique. Assume stress as an example, in a 3D space, where there nine components for stress in a cubical tiny element (3 normal and 6 shear) that can be written in matrix format.
Addition of the vectors can be done suing the polygon method or the parallelogram method which determines the resultant vector (sum of all additions). Dot product or scalar product of vector A and B shows the length (scalar) of the projection of A onto B: A.B = axbx + ayby + azbz = |A||B|Cos(θ). The cross product or vector product of vector A and B is a vector that is orthogonal to the plane that A and B make with unit vector n: A x B = |A||B|Cos(θ)n. The direction of the vector product can be determined by right-hand rule: fingers on the vector A and rotate toward B, then the Thumb shows the product of the two. There are a set of properties related to vectors that are called vector identities:
A.B = B.A
A.(B+C) = A.B + A.C
A.A = |A|2
A orthogonal to B ==> A.B = 0
AxB = -BxA
Ax(B+C) = AxB + AxC
A parallel with B ==> AxB = 0
i x j = - j x i = k
j x k = - k x j = i
k x i = - i x k = j
Nest in algebra is the subject of series. An ordered progression of numbers ai (known as terms) is called a sequence with the last term, l, known as the general term (e.g. {A} = a1, a2, a2, ..., l}. If a sequence approach toward a finite value it is a convergent sequence and if it approaches toward infinity it is divergent. When all the terms in a sequence is summed up, it is called a series. Similarly, we have finite and infinite series, though it does not mean that the sum is infinite, but the number of terms is infinite. If the sum of the terms in a series exist, it is convergent. All finite series (finite number of terms) are convergent.
We have arithmetic series is the sum of the terms in an arithmetic sequence with n terms of common difference d, (l = a + (n - 1)d), constant difference between adjacent terms. Geometric series is another standard series that converges if the common ratio, r, is between -1 and +1 and diverges otherwise. For details and examples on this type of series click here. Other types of well-known series are power series, Taylor series, and Maclaurin series (accessed 09/15/15) that are used in mathematical function derivations and integration. This page (accessed 09/15/15) also has some of the series properties.
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