Math 208 Test 1 Questions with Verified Correct
Answers
What is a linear equation in variables x₁, ..., xₙ?
It's an equation of the form a₁x₁ + a₂x₂ + ... + aₙxₙ = b, where the aᵢ's and b are real or complex
numbers, and n is a positive integer.
What is a system of linear equations?
A collection of one or more linear equations that involve the same variables.
What is a solution to a system of linear equations?
A list of values (s₁, s₂, ..., sₙ) that, when substituted into the variables (x₁, ..., xₙ), make every
equation in the system true.
What does it mean for two systems to be equivalent?
They have the same solution set.
What are the possible numbers of solutions for a system of two linear equations in two
variables?
One solution (lines intersect once)
No solution (lines are parallel)
Infinitely many solutions (lines overlap completely)
When is a system consistent or inconsistent?
Consistent: has one or infinitely many solutions
Inconsistent: has no solution
, What is the augmented matrix of a system?
A matrix that includes both the coefficients of the variables and the constants from the right-hand
side of the equations.
What are the three types of row operations?
Replacement: add a multiple of one row to another
Interchange: swap two rows
Scaling: multiply a row by a nonzero constant
What are the two fundamental questions when solving a linear system?
Is the system consistent (does a solution exist)?
If so, is the solution unique, or are there infinitely many?
What is a pivot position in a matrix?
A location that corresponds to a leading 1 in the reduced echelon form of the matrix.
What defines a matrix in row echelon form (REF)?
All nonzero rows are above rows of all zeros
Leading entry in each row is to the right of the one above it
All entries below leading entries are zeros
What additional rules define reduced row echelon form (RREF)?
Leading entry in each nonzero row is 1
Each leading 1 is the only nonzero entry in its column
What are basic and free variables?
Answers
What is a linear equation in variables x₁, ..., xₙ?
It's an equation of the form a₁x₁ + a₂x₂ + ... + aₙxₙ = b, where the aᵢ's and b are real or complex
numbers, and n is a positive integer.
What is a system of linear equations?
A collection of one or more linear equations that involve the same variables.
What is a solution to a system of linear equations?
A list of values (s₁, s₂, ..., sₙ) that, when substituted into the variables (x₁, ..., xₙ), make every
equation in the system true.
What does it mean for two systems to be equivalent?
They have the same solution set.
What are the possible numbers of solutions for a system of two linear equations in two
variables?
One solution (lines intersect once)
No solution (lines are parallel)
Infinitely many solutions (lines overlap completely)
When is a system consistent or inconsistent?
Consistent: has one or infinitely many solutions
Inconsistent: has no solution
, What is the augmented matrix of a system?
A matrix that includes both the coefficients of the variables and the constants from the right-hand
side of the equations.
What are the three types of row operations?
Replacement: add a multiple of one row to another
Interchange: swap two rows
Scaling: multiply a row by a nonzero constant
What are the two fundamental questions when solving a linear system?
Is the system consistent (does a solution exist)?
If so, is the solution unique, or are there infinitely many?
What is a pivot position in a matrix?
A location that corresponds to a leading 1 in the reduced echelon form of the matrix.
What defines a matrix in row echelon form (REF)?
All nonzero rows are above rows of all zeros
Leading entry in each row is to the right of the one above it
All entries below leading entries are zeros
What additional rules define reduced row echelon form (RREF)?
Leading entry in each nonzero row is 1
Each leading 1 is the only nonzero entry in its column
What are basic and free variables?
