LINEAR ALGEBRA EXAM GUIDE Q&A
What is a linear equation? - Answer-An equation of the form a1xa+a2x2+...+anxn=b
Where a and b are real numbers, and x is a variable
What is a system of linear equations? - Answer-A set of one or more linear equations
with the same variables
**Not all variables must appear in every equation
What is a solution? - Answer-An assignment of values to the variables that makes each
equation true
What is a solution set? - Answer-The set of all possible solutions
What makes two linear systems equivalent? - Answer-When they have the same
solution set
What types of solutions can a linear system have? - Answer-No solution, one solution,
or infinitely many solutions
What to keep track of when solving systems of linear equations - Answer-How many
variables
Which coefficients go with which variable
The "b"s
Basic strategy of solving a linear system - Answer-Replace one system with a system
that is easier to solve
What are the elementary row operations? - Answer-Replacement
Interchange
Scaling
What is replacement? - Answer-Replace a row with the sum of a nonzero multiple of
itself and a scalar multiple of another row
What is interchange? - Answer-Swap two rows
What is scaling? - Answer-Multiply a row by a nonzero constant
What does it mean to be row equivalent? - Answer-If you can get from one row to
another with elementary row operations
Are row operations reversible? - Answer-Yes
, What does it mean if augmented matrices of two linear systems are row equivalent? -
Answer-They have the same solution set
Two fundamental questions - Answer-1. Is there a solution (existence)
2. How many solutions (uniqueness)
What constitutes a matrix to be in echelon form/ row echelon form? - Answer-1. All
nonzero rows are above any rows of all zeros
2. Each leading entry is to the right of the leading entry in the row above it
3. Entries in a column below a leading entry are zero
What is reduced echelon form? - Answer-A matrix is said to be in reduced row echelon
form if...
-It is in echelon form
-The leading entry in each nonzero row is one
-Each leading one is the only nonzero entry in its column
Theorem 1 - Answer-Each matrix is row equivalent to one and one only reduced row
echelon matrix (but many echelon form matrices)
How do we define a row equivalent matrix? - Answer-If a matrix A is row equivalent to
an echelon matrix u, we call u an echelon form of A
What is a pivot? - Answer-A leading entry in a matrix in echelon form
What is a pivot position? - Answer-The locations of the pivots in echelon form
What is a pivot column? - Answer-A column containing a pivot?
What is a basic variable? - Answer-A variable corresponding to a pivot column
What is a free variable? - Answer-A variable corresponding to a column without a pivot?
What does it mean if there is a pivot in an augmented column? - Answer-No solution
What does it mean if there is not a pivot in the augmented column? - Answer-At least
one solution
What does it mean if there is a pivot in every column of the coefficient matrix? - Answer-
One solution
What does it mean if there is not a pivot in every column of the coefficient matrix? -
Answer-Infinitely many solutions
Row reduction steps - Answer-1. Begin with left most nonzero column. This is a pivot
column
What is a linear equation? - Answer-An equation of the form a1xa+a2x2+...+anxn=b
Where a and b are real numbers, and x is a variable
What is a system of linear equations? - Answer-A set of one or more linear equations
with the same variables
**Not all variables must appear in every equation
What is a solution? - Answer-An assignment of values to the variables that makes each
equation true
What is a solution set? - Answer-The set of all possible solutions
What makes two linear systems equivalent? - Answer-When they have the same
solution set
What types of solutions can a linear system have? - Answer-No solution, one solution,
or infinitely many solutions
What to keep track of when solving systems of linear equations - Answer-How many
variables
Which coefficients go with which variable
The "b"s
Basic strategy of solving a linear system - Answer-Replace one system with a system
that is easier to solve
What are the elementary row operations? - Answer-Replacement
Interchange
Scaling
What is replacement? - Answer-Replace a row with the sum of a nonzero multiple of
itself and a scalar multiple of another row
What is interchange? - Answer-Swap two rows
What is scaling? - Answer-Multiply a row by a nonzero constant
What does it mean to be row equivalent? - Answer-If you can get from one row to
another with elementary row operations
Are row operations reversible? - Answer-Yes
, What does it mean if augmented matrices of two linear systems are row equivalent? -
Answer-They have the same solution set
Two fundamental questions - Answer-1. Is there a solution (existence)
2. How many solutions (uniqueness)
What constitutes a matrix to be in echelon form/ row echelon form? - Answer-1. All
nonzero rows are above any rows of all zeros
2. Each leading entry is to the right of the leading entry in the row above it
3. Entries in a column below a leading entry are zero
What is reduced echelon form? - Answer-A matrix is said to be in reduced row echelon
form if...
-It is in echelon form
-The leading entry in each nonzero row is one
-Each leading one is the only nonzero entry in its column
Theorem 1 - Answer-Each matrix is row equivalent to one and one only reduced row
echelon matrix (but many echelon form matrices)
How do we define a row equivalent matrix? - Answer-If a matrix A is row equivalent to
an echelon matrix u, we call u an echelon form of A
What is a pivot? - Answer-A leading entry in a matrix in echelon form
What is a pivot position? - Answer-The locations of the pivots in echelon form
What is a pivot column? - Answer-A column containing a pivot?
What is a basic variable? - Answer-A variable corresponding to a pivot column
What is a free variable? - Answer-A variable corresponding to a column without a pivot?
What does it mean if there is a pivot in an augmented column? - Answer-No solution
What does it mean if there is not a pivot in the augmented column? - Answer-At least
one solution
What does it mean if there is a pivot in every column of the coefficient matrix? - Answer-
One solution
What does it mean if there is not a pivot in every column of the coefficient matrix? -
Answer-Infinitely many solutions
Row reduction steps - Answer-1. Begin with left most nonzero column. This is a pivot
column
