MA 265 EXAM 1 QUESTIONS WITH ANSWERS
What is a linear equation? - ANSWER Equation that can be written as:
a1x1 + a2x2 + ... = b
4x1 - 5x2 = x1x2 is NOT
x2 = 2(root)x1 - 6 is NOT
When is a system of linear equations consistent vs. inconsistent? - ANSWER Consistent
if system has either one or infinite solutions
Inconsistent if no solution
* a system may have no, 1, or infinite solutions
What is a coefficient matrix vs. augmented matrix? - ANSWER Coefficient matrix is when
coefficients of variables of a system of equations is expressed in a matrix
Augmented matrix is when the solutions are represented in an additional column
How many rows and columns are in a **m x n** matrix? - ANSWER M rows
N columns
When are two matrices row equivalent? - ANSWER When a sequence of elementary row
operations transforms one matrix into the other
- Row operations are reversible
- If 2 augmented matrices are row equivalent, they have the same solution set!
What are the two fundamental questions about a linear system? - ANSWER 1. Is the
system consistent / does at least one solution exist?
2. If a solution exists, is it only one / is the solution unique?
What is echelon form? What is reduced echelon form? - ANSWER Echelon:
1. All nonzero rows are above any rows of all zeros
2. Each leading entry of a row is in a column to right of leading entry of row above it
3. All entries in columns below a leading entry are zeros
Reduced echelon:
1. All of above
, 2. Leading entry in each nonzero row is 1
3. Each leading 1 is the only nonzero entry in its column
True or false: Each matrix is row equivalent to one and only one reduced echelon matrix
- ANSWER True (uniqueness)
What is a basic variable vs. free variable? - ANSWER Basic variables are expressions of
other variables and numbers, free variables can take on any value
What does the Existence and Uniqueness Theorem state? - ANSWER A linear system is
consistent if and only if the rightmost column of the augmented matrix is NOT a pivot
column
1. If system is consistent, solution contains either (1) unique solution with no free
variables or (2) infinitely many solutions with at least one free variable
What is a vector in R2? When are vectors in R2 equal? - ANSWER Vectors containing
real numbers as entries and exponent 2 means each vector has 2 entries
- 2 vectors are qual if and only if their corresponding entries are equal
- A geometric point (a,b) is column vector [a,b] (vertical ways)
A vector equation (x1a1 + x2a2 = b) has the same solution set as a linear system whose
augmented matrix is ______ - ANSWER [a1 a2 b]
What is Span{v1,...Vn}? - ANSWER Collection of all vectors that can be written in the
form of C1V1 + .. + CnVn
What does it mean to ask if vector b is in {v1,...Vn}? - ANSWER If vector equation X1V1 +
.. + XnVn = b has a solution
Let V be a nonzero vector in R3. Then Span{V} is the set of all ____ - ANSWER scalar
multiples of V, which is the set of points on the line in R3 through v and 0
If U and V are nonzero vectors in R3, with V not a multiple of U, then Span{U,V} is the
plane in ____ - ANSWER R3 that contains U, V, and 0 (plane through the origin)
The equation Ax = b has a solution if and only if b is a _____ - ANSWER linear combination
of the columns of A
If A is an MxN matrix, then the following statements are equivalent: - ANSWER 1. For
each b in Rm, the equation Ax=b has a solution
2. Each b in Rm is a linear combination of the columns of A
3. The columns of A span Rm
4. A has a pivot position in every row
What is a linear equation? - ANSWER Equation that can be written as:
a1x1 + a2x2 + ... = b
4x1 - 5x2 = x1x2 is NOT
x2 = 2(root)x1 - 6 is NOT
When is a system of linear equations consistent vs. inconsistent? - ANSWER Consistent
if system has either one or infinite solutions
Inconsistent if no solution
* a system may have no, 1, or infinite solutions
What is a coefficient matrix vs. augmented matrix? - ANSWER Coefficient matrix is when
coefficients of variables of a system of equations is expressed in a matrix
Augmented matrix is when the solutions are represented in an additional column
How many rows and columns are in a **m x n** matrix? - ANSWER M rows
N columns
When are two matrices row equivalent? - ANSWER When a sequence of elementary row
operations transforms one matrix into the other
- Row operations are reversible
- If 2 augmented matrices are row equivalent, they have the same solution set!
What are the two fundamental questions about a linear system? - ANSWER 1. Is the
system consistent / does at least one solution exist?
2. If a solution exists, is it only one / is the solution unique?
What is echelon form? What is reduced echelon form? - ANSWER Echelon:
1. All nonzero rows are above any rows of all zeros
2. Each leading entry of a row is in a column to right of leading entry of row above it
3. All entries in columns below a leading entry are zeros
Reduced echelon:
1. All of above
, 2. Leading entry in each nonzero row is 1
3. Each leading 1 is the only nonzero entry in its column
True or false: Each matrix is row equivalent to one and only one reduced echelon matrix
- ANSWER True (uniqueness)
What is a basic variable vs. free variable? - ANSWER Basic variables are expressions of
other variables and numbers, free variables can take on any value
What does the Existence and Uniqueness Theorem state? - ANSWER A linear system is
consistent if and only if the rightmost column of the augmented matrix is NOT a pivot
column
1. If system is consistent, solution contains either (1) unique solution with no free
variables or (2) infinitely many solutions with at least one free variable
What is a vector in R2? When are vectors in R2 equal? - ANSWER Vectors containing
real numbers as entries and exponent 2 means each vector has 2 entries
- 2 vectors are qual if and only if their corresponding entries are equal
- A geometric point (a,b) is column vector [a,b] (vertical ways)
A vector equation (x1a1 + x2a2 = b) has the same solution set as a linear system whose
augmented matrix is ______ - ANSWER [a1 a2 b]
What is Span{v1,...Vn}? - ANSWER Collection of all vectors that can be written in the
form of C1V1 + .. + CnVn
What does it mean to ask if vector b is in {v1,...Vn}? - ANSWER If vector equation X1V1 +
.. + XnVn = b has a solution
Let V be a nonzero vector in R3. Then Span{V} is the set of all ____ - ANSWER scalar
multiples of V, which is the set of points on the line in R3 through v and 0
If U and V are nonzero vectors in R3, with V not a multiple of U, then Span{U,V} is the
plane in ____ - ANSWER R3 that contains U, V, and 0 (plane through the origin)
The equation Ax = b has a solution if and only if b is a _____ - ANSWER linear combination
of the columns of A
If A is an MxN matrix, then the following statements are equivalent: - ANSWER 1. For
each b in Rm, the equation Ax=b has a solution
2. Each b in Rm is a linear combination of the columns of A
3. The columns of A span Rm
4. A has a pivot position in every row
