MAT1503 linear algebra (updated) exam with questions and answers (100%
solved)
matrix - ✔✔✔-A rectangular array of numbers
linear equation in n (j) unknowns - ✔✔✔-a₁x₁ + a₂x₂ + ... + aⱼxⱼ = b
linear equation - ✔✔✔-x + 3y = 7 or x₁ − 2x₂ − 3x₃ + x₄ = -1 (no products or roots of variables)
system of linear equations (linear system) - ✔✔✔-A finite set of linear equations
solution of a linear equation - ✔✔✔-A sequence of numbers for which the substitution with
variables will make the equation a true statement
homogeneous linear equations - ✔✔✔-x₁ − 2x₂ − 3x₃ + x₄ = 0
solution of a linear system - ✔✔✔-The element is a solution of each equation
solution set (general solution) - ✔✔✔-All solutions of a linear system with the number
sequence as the elements
ordered n-tuple - ✔✔✔-A linear solution written as (a₁, a₂, ... , aⱼ)
ordered pair - ✔✔✔-ordered n-tuple if n = 2
ordered triple - ✔✔✔-ordered n-tuple if n = 3
consistent system - ✔✔✔-A linear system that has at least one solution
,MAT1503 linear algebra (updated) exam with questions and answers (100%
solved)
equivalent systems - ✔✔✔-Two systems of equations that have the same solution set
inconsistent system - ✔✔✔-A linear system that has no solutions
parameter - ✔✔✔-An assigned arbitrary value where the linear system has infinite solutions
parametric equations - ✔✔✔-The solution expressed by the equations using parameters
algebraic operations - ✔✔✔-1) Add a multiple of one equation to another
2) Multiply an equation by a nonzero constant
3) Interchange two equations
Elementary Row Operations - ✔✔✔-1) Add a multiple of one row to another row
2) Multiply any row by a nonzero constant
3) Interchange two rows
augmented matrix - ✔✔✔-An abbreviation of a linear system in a rectangular array of numbers
elementary matrix - ✔✔✔-A matrix that was (or could be) produced by performing a single
Elementary Row Operation on an identity matrix
identity matrix - ✔✔✔-A square matrix with 1's on the main diagonal and zeros everywhere
else. Note A×I = A and I×A = A
Row Echelon Form - ✔✔✔-A matrix that has leading ones on the main diagonal and zeros
below the leading ones.
, MAT1503 linear algebra (updated) exam with questions and answers (100%
solved)
Reduced Row Echelon Form - ✔✔✔-A matrix that has leading ones on the main diagonal and
zeros above and below the leading ones.
leading variables - ✔✔✔-The variables corresponding to the leading 1's in the augmented
matrix
free variables - ✔✔✔-The variables that can be assigned an arbitrary value
Gaussian Elimination - ✔✔✔-1) Put the matrix in augmented matrix form
2) Use row operations to put the matrix in echelon form
3) Write the equations from the echelon form matrix
4) Solve the equations.
Gauss-Jordan Elimination - ✔✔✔-1) Put the matrix in augmented matrix form
2) Use row operations to put the matrix in reduced echelon form
3) Write the equations from the echelon form matrix
4) Solve the equations.
trivial solution - ✔✔✔-The solutions of the homogeneous linear systems are 0
non-trivial solution - ✔✔✔-The solutions of the homogeneous linear systems are infinite (free
variables are used)
Free Variable Theorem for Homogeneous Systems - ✔✔✔-If a homogeneous linear system has
n unknowns, and its augmented matrix has r nonzero rows in reduced row echelon form, then
the system has n - r free variables
solved)
matrix - ✔✔✔-A rectangular array of numbers
linear equation in n (j) unknowns - ✔✔✔-a₁x₁ + a₂x₂ + ... + aⱼxⱼ = b
linear equation - ✔✔✔-x + 3y = 7 or x₁ − 2x₂ − 3x₃ + x₄ = -1 (no products or roots of variables)
system of linear equations (linear system) - ✔✔✔-A finite set of linear equations
solution of a linear equation - ✔✔✔-A sequence of numbers for which the substitution with
variables will make the equation a true statement
homogeneous linear equations - ✔✔✔-x₁ − 2x₂ − 3x₃ + x₄ = 0
solution of a linear system - ✔✔✔-The element is a solution of each equation
solution set (general solution) - ✔✔✔-All solutions of a linear system with the number
sequence as the elements
ordered n-tuple - ✔✔✔-A linear solution written as (a₁, a₂, ... , aⱼ)
ordered pair - ✔✔✔-ordered n-tuple if n = 2
ordered triple - ✔✔✔-ordered n-tuple if n = 3
consistent system - ✔✔✔-A linear system that has at least one solution
,MAT1503 linear algebra (updated) exam with questions and answers (100%
solved)
equivalent systems - ✔✔✔-Two systems of equations that have the same solution set
inconsistent system - ✔✔✔-A linear system that has no solutions
parameter - ✔✔✔-An assigned arbitrary value where the linear system has infinite solutions
parametric equations - ✔✔✔-The solution expressed by the equations using parameters
algebraic operations - ✔✔✔-1) Add a multiple of one equation to another
2) Multiply an equation by a nonzero constant
3) Interchange two equations
Elementary Row Operations - ✔✔✔-1) Add a multiple of one row to another row
2) Multiply any row by a nonzero constant
3) Interchange two rows
augmented matrix - ✔✔✔-An abbreviation of a linear system in a rectangular array of numbers
elementary matrix - ✔✔✔-A matrix that was (or could be) produced by performing a single
Elementary Row Operation on an identity matrix
identity matrix - ✔✔✔-A square matrix with 1's on the main diagonal and zeros everywhere
else. Note A×I = A and I×A = A
Row Echelon Form - ✔✔✔-A matrix that has leading ones on the main diagonal and zeros
below the leading ones.
, MAT1503 linear algebra (updated) exam with questions and answers (100%
solved)
Reduced Row Echelon Form - ✔✔✔-A matrix that has leading ones on the main diagonal and
zeros above and below the leading ones.
leading variables - ✔✔✔-The variables corresponding to the leading 1's in the augmented
matrix
free variables - ✔✔✔-The variables that can be assigned an arbitrary value
Gaussian Elimination - ✔✔✔-1) Put the matrix in augmented matrix form
2) Use row operations to put the matrix in echelon form
3) Write the equations from the echelon form matrix
4) Solve the equations.
Gauss-Jordan Elimination - ✔✔✔-1) Put the matrix in augmented matrix form
2) Use row operations to put the matrix in reduced echelon form
3) Write the equations from the echelon form matrix
4) Solve the equations.
trivial solution - ✔✔✔-The solutions of the homogeneous linear systems are 0
non-trivial solution - ✔✔✔-The solutions of the homogeneous linear systems are infinite (free
variables are used)
Free Variable Theorem for Homogeneous Systems - ✔✔✔-If a homogeneous linear system has
n unknowns, and its augmented matrix has r nonzero rows in reduced row echelon form, then
the system has n - r free variables
