egm4313 exam1 review statics

8/13/2019 Egm4313 Exam1 Review Statics

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University of West Florida

Department of Electrical and Computer Engineering

EGM4313 Intermediate Engineering Analysis

Review Material for Exam 1

Exam #1, which will be on Thursday, 30 January, will cover material from chapter 7 of the coursetext that is associated with lectures. An additional topic, which is not discussed in the course text,

is the determinination of solutions to underdetermined or overdetermined systems using generalized

inverses.

The below list includes topics that should be understood for the first exam. The list is intended to

be complete but is not necessarily all inclusive.

Understand general concepts associated with scalars, vectors and matrices.

The sum/difference of matrices is only defined if all matrices have the same same di-mensions. That is, each matrix must have n rows and m columns. The sum/difference

is the sum/difference on a component by component basis.

The product of matrices is only defined if the inner dimension of each product is the

same. That is, for each product term C= AB , the number of columns of Amust equal

the number of rows of B and the dimension of C is such that the rows are equal to the

rows of A and the columns are equal to the columns of B. The product is obtained in

such a way that the term in the ith row andj th column of Cis the inner product of the

ith row of Aand the jth column of B.

Multiplication of a matrix by a scalar is a special case and is obtained by multiplyingeach element of the matrix by the scalar.

Matrix multiplication is generally not commutative so AB= B A.

Many special square matrices exist.

A diagonal matrix A has all elements not on the main diagonal equal to zero (i.e.,

aij = 0, i = j) while all diagonal elements (i.e., aij, i = j) are in general nonzero.

The identity matrix, I, is a special diagonal matrix in that all diagonal elements

are 1.

An upper (lower) triangular matrix has all elements below (above) the main diagonal

equal to 0. All other elements in general are not equal to 0.

The solution to the linear equation Ax= b where A is the nn coefficient matrix, x is an

n1 unknown vector and b is an 1 known vector is obtained by elementary row operations

of the augmented matrix

A b

R

A b

.

EGM4313 1 Spring 2014 Exam #1 Review


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If no rows in A are entirely 0, then the system has a unique solution.

If one (or more) rows in A are entirely 0 and the corresponding row(s) in b are zero,

then the system has no unique solution (i.e., there are an infinite number of solutions).

If at least one row in A is entirely 0 and the corresponding row in b is not zero, then

the system has no solution (i.e., there are inconsistencies in the original equations).

Valid row operations consist of multiplying a row by a nonzero constant, adding a scaled

version of one row to some other row and interchanging rows.

To determine linear independence of row (column) vectors, arrange the row (column) vectors

in row (column) order in a matrix.

Using elementary row operations, determine the number of rows that are not entirely 0.

This number indicates the number of linearly independent rows. Note that the number

of linearly independent rows must be less than or equal to the smallest dimension of the

matrix. Further, the number of linearly independent rows must equal the number of

linearly independent columns.

If the number rows is equal to the number of elements in the row, then the resulting

matrix is square. As such, one can utilize the determinant to ascertain linear indepen-

dence.

Rows are linearly independent provided that the determinant of the matrix is nonzero.

If any row of a matrix is zero or is a linear combination of the other rows, then

the determinant will be zero.

If any two rows of the matrix are interchanged, then the value of the determinant

will be multiplied by 1.

Adding a nonzero scalar multiple of one row to another row does not change the

value of the determinant.

The determinant of a diagonal or upper (lower) triangular matrix is the product

of the diagonal elements.

The inverse of a square matrix A is found by augmenting A with the identity matrix and

then applying elementary row operations to both Aand Iuntil the original Ais the identity

matrix which implies that the orignial identity matrix is now A1 or the inverse of A.

A I R

I A1

If A is a diagonal matrix, then A1 is a diagonal matrix with diagonal elements that

are reciprocals of the ones in A.

Under the condition that all the matrices A, B, . . . , Y Zare nn, if

A = BC Y Z,



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then

A1 = (BC Y Z)1 = Z1Y1 C1B1.

For the overspecified linear system Ax= b where Ais m nwith m > nand rank{A}= n,

bis m1 and x is n1, the minimum norm solution (i.e., it is not an exact solution) is

x = (AA)1 Ab,

where A is the complex conjugate transpose of A.

The transpose of a matrix Ais obtained by interchanging rows of Awith columns of A.

The complex conjugate transpose simply replaces all complex terms with their conjugate

after the transpose has been found.

For the underspecified linear system Ax= b where A ismnwithm < nand rank{A}= m,

bis m 1 and xis n 1, a solution (i.e., there are an infinite number of exact solutions; the

one found minimizes the length of x) is

x = A (AA)1 b,

where A is the complex conjugate transpose of A.


egm4313 exam1 review statics

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