UMBC MATH 221: Exam 1 Questions with
Verified Correct Answers
Define:
Equivalent
When two or more systems have the same solution set
Define:
Inconsistent
When a linear system has no solution
Define:
Consistent
When a linear system has at least one solution
True or False:
If a matrix is mxn, it has m rows and n columns
True
What are the two fundamental questions to ask about a linear system?
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):
Each matrix is row equivalent to ___ and only ___ reduced row echelon matrix.
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)
- 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
A vector that is the sum of other vectors that are multiplied by scalars
Ex. y (linear combination) = c₁v₁ + c₂v₂ + ... +cⁿvⁿ
Define:
Span
The subset of Rⁿ that contains all linear combinations of the vectors v¹, v², ..., vⁿ
- Sp{v¹ ... vⁿ} = c¹v¹ + ... + cⁿvⁿ
What does Ax denote?
The multiplication of a matrix, A, by the vector, x.
Ex.
A = [a¹ a² ... aⁿ]
x = [x¹
x²
Verified Correct Answers
Define:
Equivalent
When two or more systems have the same solution set
Define:
Inconsistent
When a linear system has no solution
Define:
Consistent
When a linear system has at least one solution
True or False:
If a matrix is mxn, it has m rows and n columns
True
What are the two fundamental questions to ask about a linear system?
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):
Each matrix is row equivalent to ___ and only ___ reduced row echelon matrix.
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)
- 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
A vector that is the sum of other vectors that are multiplied by scalars
Ex. y (linear combination) = c₁v₁ + c₂v₂ + ... +cⁿvⁿ
Define:
Span
The subset of Rⁿ that contains all linear combinations of the vectors v¹, v², ..., vⁿ
- Sp{v¹ ... vⁿ} = c¹v¹ + ... + cⁿvⁿ
What does Ax denote?
The multiplication of a matrix, A, by the vector, x.
Ex.
A = [a¹ a² ... aⁿ]
x = [x¹
x²
