UMBC MATH 221: EXAM 1 QUESTIONS
AND ANSWERS. VERIFIED 2026.
Define:
Equivalent - ANS When two or more systems have the same solution set
Define:
Inconsistent - ANS When a linear system has no solution
Define:
Consistent - ANS When a linear system has at least one solution
True or False:
If a matrix is mxn, it has m rows and n columns - ANS True
What are the two fundamental questions to ask about a linear system? - ANS 1) Is the system
consistent? (Does at least one solution exist?)
2) If a solution exists, is it the only one? (Is the solution unique?)
Fill in the blanks from Theorem 1 (Ch. 1):
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1
, Each matrix is row equivalent to ___ and only ___ reduced row echelon matrix. - ANS Each
matrix is row equivalent to ONE and only ONE reduced row echelon matrix.
Fill in the blanks from Theorem 2 (Ch. 1):
- A linear system will not be consistent if the ___most column of the augmented matrix is a
pivot column
Ex. If the matrix has a row [ _ ... _ _ b ] where b =/= 0, it is inconsistent
- A consistent linear system either has one unique solution (___free variables) or infinitely many
solutions (with at least ____ free variable) - ANS - A linear system will not be consistent if the
RIGHTmost column of the augmented matrix is a pivot column
Ex. If the matrix has a row [0 ...0 0 b] where b =/= 0, it is inconsistent
- A consistent linear system either has one unique solution (NO free variables) or infinitely many
solutions (with at least ONE free variable)
Define:
Linear combination - ANS A vector that is the sum of other vectors that are multiplied by
scalars
Ex. y (linear combination) = c₁v₁ + c₂v₂ + ... +cⁿvⁿ
Define:
Span - ANS The subset of Rⁿ that contains all linear combinations of the vectors v¹, v², ..., vⁿ
- Sp{v¹ ... vⁿ} = c¹v¹ + ... + cⁿvⁿ
What does Ax denote? - ANS The multiplication of a matrix, A, by the vector, x.
Ex.
A = [a¹ a² ... aⁿ]
x = [x¹
@COPYRIGHT 2026/2027 ALL RIGHTS RESERVED
2
AND ANSWERS. VERIFIED 2026.
Define:
Equivalent - ANS When two or more systems have the same solution set
Define:
Inconsistent - ANS When a linear system has no solution
Define:
Consistent - ANS When a linear system has at least one solution
True or False:
If a matrix is mxn, it has m rows and n columns - ANS True
What are the two fundamental questions to ask about a linear system? - ANS 1) Is the system
consistent? (Does at least one solution exist?)
2) If a solution exists, is it the only one? (Is the solution unique?)
Fill in the blanks from Theorem 1 (Ch. 1):
@COPYRIGHT 2026/2027 ALL RIGHTS RESERVED
1
, Each matrix is row equivalent to ___ and only ___ reduced row echelon matrix. - ANS Each
matrix is row equivalent to ONE and only ONE reduced row echelon matrix.
Fill in the blanks from Theorem 2 (Ch. 1):
- A linear system will not be consistent if the ___most column of the augmented matrix is a
pivot column
Ex. If the matrix has a row [ _ ... _ _ b ] where b =/= 0, it is inconsistent
- A consistent linear system either has one unique solution (___free variables) or infinitely many
solutions (with at least ____ free variable) - ANS - A linear system will not be consistent if the
RIGHTmost column of the augmented matrix is a pivot column
Ex. If the matrix has a row [0 ...0 0 b] where b =/= 0, it is inconsistent
- A consistent linear system either has one unique solution (NO free variables) or infinitely many
solutions (with at least ONE free variable)
Define:
Linear combination - ANS A vector that is the sum of other vectors that are multiplied by
scalars
Ex. y (linear combination) = c₁v₁ + c₂v₂ + ... +cⁿvⁿ
Define:
Span - ANS The subset of Rⁿ that contains all linear combinations of the vectors v¹, v², ..., vⁿ
- Sp{v¹ ... vⁿ} = c¹v¹ + ... + cⁿvⁿ
What does Ax denote? - ANS The multiplication of a matrix, A, by the vector, x.
Ex.
A = [a¹ a² ... aⁿ]
x = [x¹
@COPYRIGHT 2026/2027 ALL RIGHTS RESERVED
2
