Linear Algebra Ch.1

Front 1

A linear equation in n variables x1, x2, x3, ... , xn has the form __________.

Back 1

a1x1 + a2x2 + a3x3 + ... + anxn = b

Front 2

A systems of linear equations in which each of the constant terms is zero is called ________.

Back 2

Homogeneous

Front 3

Reduced row echelon form (rref):

Back 3

1. all rows consisting entirely of zeros at bottom
2. the first nonzero entry in any nonzero row is 1 (called leading 1)
3. the leading in each nonzero row is at least one column to right of leading 1 in previous row
4. all entries ABOVE (and below) any leading 1 are zeros

Front 4

To solve a system in ref you use a procedure called __________.

Back 4

back substitution

Front 5

Two systems of linear equations are called ________ if they have precisely the same solution set.

Back 5

Equivalent

Front 6

Operations that lead to equivalent systems of equations. For a system of linear equations in n variables, precisely one of the following is true:

Back 6

1. The system has exactly one solution (consistent system).
2. The system has an infinite number of solutions (consistent system).
3. The system has no solution (inconsistent system).

Front 7

The number a1 is the ________ coefficient.

Back 7

Leading

Front 8

Elementary Row Operations (ERO):

Back 8

Elementary Row Operations (ERO):
1. Interchange two rows.
2. Multiply a row by a nonzero constant.
3. Add a multiple of a row to another row.

Front 9

If all variables in a homogeneous system have the value zero, then each of the equations must be satisfied. Such a solution is called _______ (or obvious).

Back 9

Trivial

Front 10

An m*n matrix (read "m by n") has m _____ (horizontal lines) and n _______ (vertical lines).

Back 10

rows, columns

Front 11

If m and n are positive integers, then an m*n ______ is a rectangular array in which each entry, aij, of the matrix is a number.

Back 11

Matrix

Front 12

A homogeneous system must have at least ___________.

Back 12

one solution

Front 13

A system of linear equations is called _________ if it has at least one solution and _________ if it has no solution.

Back 13

consistent, inconsistent

Front 14

A matrix in row-echelon form (ref) has the following properties:

Back 14

1. All rows consisting entirely of zeros occur at the bottom of the matrix.
2. For each row that does not consist entirely of zeros, the first nonzero entry is 1 (called a leading 1).
3. For two succesive (nonzero) rows, the leading 1 in the higher row is farther to the left than the leading 1 in the lower row.

Front 15

x1 is the leading _______.

Back 15

Variable

Front 16

For a system of linear equations in n variables, precisely one of the following is true:

Back 16

1. The system has exactly one solution (consistent system).
2. The system has an infinite number of solutions (consistent system).
3. The system has no solution (inconsistent system).

Front 17

The ______ term b is a real number.

Back 17

Constant

Front 18

The real numbers (a1, a2, a3, ... , an) are ________.

Back 18

Coefficients

Front 19

Row-echelon form (ref) means:

Back 19

the system has leading coefficients of 1 that follows a stair-step pattern

example:
1 2 3 9
0 1 3 5
0 0 1 2

Front 20

The set of all solutions of a linear equations is called its __________, and when this set is found, the equation is said to have been __________.

Back 20

solution set, solved

Front 21

Every homogeneous system of linear equations is __________. Moreover, if the system has fewer equations than variables, then it must have an __________ number of solutions.

Back 21

consistent, infinite