MATH 225 UIUC
Midterm 1 Theorems
with complete verified
solutions 2025-2026
A system of linear equations (=SLE or a linear system): - answer A finite
collection of one or more linear equations involving the
same set of variables, say, x1, x2,..., xn.
A solution of a SLE, - answer A list s1,s2,...,sn of numbers that makes each
equation in the
system true when the values s1,s2,...,sn are substituted for
x1, x2,..., xn, respectively.
A system of linear equations has - answer (i) exactly one solution
(consistent) or
(ii) infinitely many solutions (consistent) or
(iii) no solution (inconsistent).
Two Fundamental Questions (Existence and Uniqueness) - answer 1) Is the
system consistent? (i.e. does a solution exist?)
2) If a solution exists, is it unique? (i.e. is there one & only one
solution?)
Row Echelon Form - answer 1. All nonzero rows are above any rows of all
zeros, called 0-rows.
2. Each leading entry (i.e. left most nonzero 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 zero
Pivot position: - answer a position of a leading entry in an echelon
form of the matrix.
Pivot - answer a nonzero number that either is used in a pivot
position to create 0's or is changed into a leading 1 which in
turn is used to create 0's.
pivot column - answer a column that contains a pivot position.
infinitely many solutions - answer consistent system with free variables
unique solution - answer consistent system, no free variables
Theorem 2 on Existence and Uniqueness of Solutions - answer 1. A linear
system is consistent if and only if the rightmost
column of the augmented matrix is not a pivot column, i.e., if
and only if a row echelon form of the augmented matrix has no
row of the form [0...0 b] where b is nonzero
2. If a linear system is consistent, then the solution set
contains either
(i) a unique solution (when there are no free variables) or
(ii) infinitely many solutions (when there is at least one free
variable).
algorithm to solve a linear system - answer 1. Write down the augmented
matrix of given SLE.
2. Use the row reduction algorithm to obtain a row equivalent
Midterm 1 Theorems
with complete verified
solutions 2025-2026
A system of linear equations (=SLE or a linear system): - answer A finite
collection of one or more linear equations involving the
same set of variables, say, x1, x2,..., xn.
A solution of a SLE, - answer A list s1,s2,...,sn of numbers that makes each
equation in the
system true when the values s1,s2,...,sn are substituted for
x1, x2,..., xn, respectively.
A system of linear equations has - answer (i) exactly one solution
(consistent) or
(ii) infinitely many solutions (consistent) or
(iii) no solution (inconsistent).
Two Fundamental Questions (Existence and Uniqueness) - answer 1) Is the
system consistent? (i.e. does a solution exist?)
2) If a solution exists, is it unique? (i.e. is there one & only one
solution?)
Row Echelon Form - answer 1. All nonzero rows are above any rows of all
zeros, called 0-rows.
2. Each leading entry (i.e. left most nonzero 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 zero
Pivot position: - answer a position of a leading entry in an echelon
form of the matrix.
Pivot - answer a nonzero number that either is used in a pivot
position to create 0's or is changed into a leading 1 which in
turn is used to create 0's.
pivot column - answer a column that contains a pivot position.
infinitely many solutions - answer consistent system with free variables
unique solution - answer consistent system, no free variables
Theorem 2 on Existence and Uniqueness of Solutions - answer 1. A linear
system is consistent if and only if the rightmost
column of the augmented matrix is not a pivot column, i.e., if
and only if a row echelon form of the augmented matrix has no
row of the form [0...0 b] where b is nonzero
2. If a linear system is consistent, then the solution set
contains either
(i) a unique solution (when there are no free variables) or
(ii) infinitely many solutions (when there is at least one free
variable).
algorithm to solve a linear system - answer 1. Write down the augmented
matrix of given SLE.
2. Use the row reduction algorithm to obtain a row equivalent
