Linear Algebra Final exam actual questions
and answers.
A rectangle matrix is in echelon form if it has the following properties:
1. All nonzero rows are above any rows of all zero.
2. Each leading entry of a row is in a column to the right of the leading
entry of the row above it.
3. All entries in a column below a leading entry are zeros.
If a matrix in echelon form satisfies the following additional conditions,
then it is in reduced echelon form.
1. The leading entry in each nonzero row is 1
2. Each leading 1 is the only nonzero entry in its column.
Pivot position
A location in a matrix A that corresponds to a leading 1 in the reduced
echelon form of A.
Pivot column
A column of A that contains a pivot position
Basic variable
1|Page
,Variables corresponding to pivot columns (AKA leading variables)
Free variable
Variables that DO NOT correspond to a pivot position. Their value can
influence the rest of the basic variables.
General solution
Gives an explicit definition of all the solutions
Consistent
A matrix is _________ if and only if the rightmost column of the
augmented matrix is not a pivot column
Unique solution
A solution in which there are no free variables
Infinitely many solutions
A solution in which there is at least one free variable
Column vector
A matrix with only one column
2|Page
,If u and v in R^2 are represented as points in the plane, then u + v
corresponds to the ____________of the parallelogram whose other
vertices are u, 0 , and v
fourth vertex
Let A be an nxn matrix. Then A is invertible if and only if:
What is the subset of R^n spanned or generated by v1, v2, ..., vp?
The set of all linear combinations of v1, v2, .... vp denoted by Span{v1,
v2,...., vp} that can be written in the form c1v1 + c2v2, + ..... cpvp.
What is the Span of something geometrically?
A line through the origin
If u and v are nonzero vectors in R^3, with v not a multiple of u, what is
the Span of u and v geometrically?
A plane in R^3 that contains u, v, and 0
If A is an m x n matrix, with columns a1, a2, ..., an, and if x is in R^n,
then the product of A an dx , denoted by Ax, is what?
the linear combination of the columns of A using the corresponding
entries in x as weights
3|Page
, If A is an m x n matrix, with columns a1, a2, ...., an, and if b is in R^m,
the ____________________ has the same solution set as the
___________, which in turn has the same solution set as the system of
linear equations whose _________________________ is
_____________.
matrix equation, Ax = b;
vector equation, x1a1 + x2a2 + ..... + xnan = b
augmented matrix, [a1, a2, .... an b]
Let A be an n x n matrix Then A is invertible if and only if:
- The number 0 is not an eigenvalue of A.
- The determinant of A is not zero
What does the absolute value of the determinant become when a matrix
is a 3 x 3 matrix?
The volume of the parallelpiped determined by columns a1, a2, a3
The equation Ax = b has a solution if and only if
____________________________.
b is a linear combination of the columns of A
What do we mean when we say the columns of A span R^m?
Every b in R^m is a linear combination of the columns of A
4|Page
and answers.
A rectangle matrix is in echelon form if it has the following properties:
1. All nonzero rows are above any rows of all zero.
2. Each leading entry of a row is in a column to the right of the leading
entry of the row above it.
3. All entries in a column below a leading entry are zeros.
If a matrix in echelon form satisfies the following additional conditions,
then it is in reduced echelon form.
1. The leading entry in each nonzero row is 1
2. Each leading 1 is the only nonzero entry in its column.
Pivot position
A location in a matrix A that corresponds to a leading 1 in the reduced
echelon form of A.
Pivot column
A column of A that contains a pivot position
Basic variable
1|Page
,Variables corresponding to pivot columns (AKA leading variables)
Free variable
Variables that DO NOT correspond to a pivot position. Their value can
influence the rest of the basic variables.
General solution
Gives an explicit definition of all the solutions
Consistent
A matrix is _________ if and only if the rightmost column of the
augmented matrix is not a pivot column
Unique solution
A solution in which there are no free variables
Infinitely many solutions
A solution in which there is at least one free variable
Column vector
A matrix with only one column
2|Page
,If u and v in R^2 are represented as points in the plane, then u + v
corresponds to the ____________of the parallelogram whose other
vertices are u, 0 , and v
fourth vertex
Let A be an nxn matrix. Then A is invertible if and only if:
What is the subset of R^n spanned or generated by v1, v2, ..., vp?
The set of all linear combinations of v1, v2, .... vp denoted by Span{v1,
v2,...., vp} that can be written in the form c1v1 + c2v2, + ..... cpvp.
What is the Span of something geometrically?
A line through the origin
If u and v are nonzero vectors in R^3, with v not a multiple of u, what is
the Span of u and v geometrically?
A plane in R^3 that contains u, v, and 0
If A is an m x n matrix, with columns a1, a2, ..., an, and if x is in R^n,
then the product of A an dx , denoted by Ax, is what?
the linear combination of the columns of A using the corresponding
entries in x as weights
3|Page
, If A is an m x n matrix, with columns a1, a2, ...., an, and if b is in R^m,
the ____________________ has the same solution set as the
___________, which in turn has the same solution set as the system of
linear equations whose _________________________ is
_____________.
matrix equation, Ax = b;
vector equation, x1a1 + x2a2 + ..... + xnan = b
augmented matrix, [a1, a2, .... an b]
Let A be an n x n matrix Then A is invertible if and only if:
- The number 0 is not an eigenvalue of A.
- The determinant of A is not zero
What does the absolute value of the determinant become when a matrix
is a 3 x 3 matrix?
The volume of the parallelpiped determined by columns a1, a2, a3
The equation Ax = b has a solution if and only if
____________________________.
b is a linear combination of the columns of A
What do we mean when we say the columns of A span R^m?
Every b in R^m is a linear combination of the columns of A
4|Page
