MAT 223 DEFINITION EXAM QUESTIONS WITH VERIFIED ANSWERS
LATEST UPDATE 2026
Linear Equation - Answers An equation of the form a₁x₁ + a₂x₂ + ··· + aₙxₙ = b in
variables x₁, x₂, ..., xₙ, where a₁, ..., aₙ and b are real numbers. (Definition 1.1)
System of Linear Equations - Answers A collection of one or more linear equations
in the same variables. (Definition 1.3)
Solution to a System of Linear Equations - Answers A tuple (s₁, ..., sₙ) ∈ ℝⁿ that
satisfies every linear equation in the system simultaneously. (Definition 1.3)
Matrix - Answers Any rectangular array of quantities or expressions (called its
entries). If it has m rows and n columns, it is an m × n matrix. (Definition 1.5)
Entries of a Matrix - Answers The individual quantities or expressions positioned
inside a rectangular array/matrix. (Definition 1.5)
Coefficient Matrix - Answers The m × n matrix containing only the coefficients of
the variables from a system of m linear equations in n variables. (Definition 1.6)
Augmented Matrix - Answers The m × (n + 1) matrix formed by adjoining the
constant column vector on the right-hand side of a linear system to its coefficient
matrix. (Definition 1.6)
Equivalent Systems - Answers Two systems of linear equations that have the exact
same solution set. (Definition 1.9)
Elementary Row Operations - Answers Three allowable row operations: (ER1)
Interchange two rows; (ER2) Multiply a row by a non-zero scalar; (ER3) Replace a
row by the sum of that row and a scalar multiple of another row. (Definition 1.11)
Row Equivalent Matrices (A ~ B) - Answers Two matrices A and B where one can
be transformed into the other via a finite sequence of elementary row operations.
(Definition 1.11)
Pivot of a Row - Answers The leftmost non-zero entry in a non-zero row of a matrix.
(Definition 1.13)
Row Echelon Form (REF) - Answers A matrix state where: (1) all rows consisting
entirely of zeros are at the bottom, and (2) the pivot of each non-zero row lies in a
column strictly to the right of the pivot of the row above it. (Definition 1.15)
Reduced Row Echelon Form (RREF) - Answers A matrix in REF where additionally:
(1) every pivot equals 1, and (2) each pivot is the only non-zero entry in its entire
column. (Definition 1.16)
, Reduced Row Echelon Form of A (rref(A)) - Answers The unique matrix in reduced
row echelon form that is row equivalent to A. (Definition 1.21)
Consistent System - Answers A system of linear equations that has at least one
solution (either a unique solution or infinitely many solutions). (Definition 1.23)
Inconsistent System - Answers A system of linear equations that has no solutions
(corresponds to a pivot in the augmented column of rref(A)). (Definition 1.23)
Basic Variable - Answers A variable xᵢ whose corresponding column i in rref(C) of
the coefficient matrix contains a pivot. (Definition 1.24)
Free Variable - Answers A variable xᵢ whose corresponding column i in rref(C) of the
coefficient matrix does not contain a pivot; it can take on any real parameter.
(Definition 1.24)
n-Dimensional Vector - Answers An ordered list of n real numbers, written as a
column vector with standard coordinates v₁, v₂, ..., vₙ. (Definition 2.3)
Standard Coordinates of a Vector - Answers The ordered list of scalar components
v₁, v₂, ..., vₙ that uniquely determine a vector in ℝⁿ. (Definition 2.3)
n-Dimensional Euclidean Space (ℝⁿ) - Answers The set of all n-dimensional column
vectors whose entries are real numbers. (Definition 2.4)
Vector Addition in ℝⁿ - Answers The component-wise sum of two vectors: [v₁ + w₁,
v₂ + w₂, ..., vₙ + wₙ]ᵀ. (Definition 2.5)
Scalar Multiplication in ℝⁿ - Answers The operation of multiplying every component
of a vector by a real scalar c: [cv₁, cv₂, ..., cvₙ]ᵀ. (Definition 2.5)
Linear Combination - Answers A vector of the form c₁v₁ + c₂v₂ + ··· + cₙvₙ, where
c₁, ..., cₙ are real scalars called coefficients. (Definition 2.8)
Span (Span(v₁, ..., vₙ)) - Answers The set of all possible linear combinations of the
vectors v₁, v₂, ..., vₙ: {c₁v₁ + ··· + cₙvₙ | cᵢ ∈ ℝ}. (Definition 2.9)
Linearly Dependent Set - Answers A set of vectors {v₁, ..., vₙ} where at least one
vector can be written as a linear combination of the remaining vectors. (Definition
2.10)
Linearly Independent Set - Answers A set of vectors {v₁, ..., vₙ} where no vector can
be written as a linear combination of the others (equivalently, c₁v₁ + ··· + cₙvₙ = 0
only has the trivial solution c₁ = ··· = cₙ = 0). (Definition 2.10)
Vector Space (over ℝ) - Answers A non-empty subset V ⊆ ℝⁿ closed under vector
addition (v + w ∈ V) and scalar multiplication (cv ∈ V). (Definition 3.1)
LATEST UPDATE 2026
Linear Equation - Answers An equation of the form a₁x₁ + a₂x₂ + ··· + aₙxₙ = b in
variables x₁, x₂, ..., xₙ, where a₁, ..., aₙ and b are real numbers. (Definition 1.1)
System of Linear Equations - Answers A collection of one or more linear equations
in the same variables. (Definition 1.3)
Solution to a System of Linear Equations - Answers A tuple (s₁, ..., sₙ) ∈ ℝⁿ that
satisfies every linear equation in the system simultaneously. (Definition 1.3)
Matrix - Answers Any rectangular array of quantities or expressions (called its
entries). If it has m rows and n columns, it is an m × n matrix. (Definition 1.5)
Entries of a Matrix - Answers The individual quantities or expressions positioned
inside a rectangular array/matrix. (Definition 1.5)
Coefficient Matrix - Answers The m × n matrix containing only the coefficients of
the variables from a system of m linear equations in n variables. (Definition 1.6)
Augmented Matrix - Answers The m × (n + 1) matrix formed by adjoining the
constant column vector on the right-hand side of a linear system to its coefficient
matrix. (Definition 1.6)
Equivalent Systems - Answers Two systems of linear equations that have the exact
same solution set. (Definition 1.9)
Elementary Row Operations - Answers Three allowable row operations: (ER1)
Interchange two rows; (ER2) Multiply a row by a non-zero scalar; (ER3) Replace a
row by the sum of that row and a scalar multiple of another row. (Definition 1.11)
Row Equivalent Matrices (A ~ B) - Answers Two matrices A and B where one can
be transformed into the other via a finite sequence of elementary row operations.
(Definition 1.11)
Pivot of a Row - Answers The leftmost non-zero entry in a non-zero row of a matrix.
(Definition 1.13)
Row Echelon Form (REF) - Answers A matrix state where: (1) all rows consisting
entirely of zeros are at the bottom, and (2) the pivot of each non-zero row lies in a
column strictly to the right of the pivot of the row above it. (Definition 1.15)
Reduced Row Echelon Form (RREF) - Answers A matrix in REF where additionally:
(1) every pivot equals 1, and (2) each pivot is the only non-zero entry in its entire
column. (Definition 1.16)
, Reduced Row Echelon Form of A (rref(A)) - Answers The unique matrix in reduced
row echelon form that is row equivalent to A. (Definition 1.21)
Consistent System - Answers A system of linear equations that has at least one
solution (either a unique solution or infinitely many solutions). (Definition 1.23)
Inconsistent System - Answers A system of linear equations that has no solutions
(corresponds to a pivot in the augmented column of rref(A)). (Definition 1.23)
Basic Variable - Answers A variable xᵢ whose corresponding column i in rref(C) of
the coefficient matrix contains a pivot. (Definition 1.24)
Free Variable - Answers A variable xᵢ whose corresponding column i in rref(C) of the
coefficient matrix does not contain a pivot; it can take on any real parameter.
(Definition 1.24)
n-Dimensional Vector - Answers An ordered list of n real numbers, written as a
column vector with standard coordinates v₁, v₂, ..., vₙ. (Definition 2.3)
Standard Coordinates of a Vector - Answers The ordered list of scalar components
v₁, v₂, ..., vₙ that uniquely determine a vector in ℝⁿ. (Definition 2.3)
n-Dimensional Euclidean Space (ℝⁿ) - Answers The set of all n-dimensional column
vectors whose entries are real numbers. (Definition 2.4)
Vector Addition in ℝⁿ - Answers The component-wise sum of two vectors: [v₁ + w₁,
v₂ + w₂, ..., vₙ + wₙ]ᵀ. (Definition 2.5)
Scalar Multiplication in ℝⁿ - Answers The operation of multiplying every component
of a vector by a real scalar c: [cv₁, cv₂, ..., cvₙ]ᵀ. (Definition 2.5)
Linear Combination - Answers A vector of the form c₁v₁ + c₂v₂ + ··· + cₙvₙ, where
c₁, ..., cₙ are real scalars called coefficients. (Definition 2.8)
Span (Span(v₁, ..., vₙ)) - Answers The set of all possible linear combinations of the
vectors v₁, v₂, ..., vₙ: {c₁v₁ + ··· + cₙvₙ | cᵢ ∈ ℝ}. (Definition 2.9)
Linearly Dependent Set - Answers A set of vectors {v₁, ..., vₙ} where at least one
vector can be written as a linear combination of the remaining vectors. (Definition
2.10)
Linearly Independent Set - Answers A set of vectors {v₁, ..., vₙ} where no vector can
be written as a linear combination of the others (equivalently, c₁v₁ + ··· + cₙvₙ = 0
only has the trivial solution c₁ = ··· = cₙ = 0). (Definition 2.10)
Vector Space (over ℝ) - Answers A non-empty subset V ⊆ ℝⁿ closed under vector
addition (v + w ∈ V) and scalar multiplication (cv ∈ V). (Definition 3.1)
