LINEAR ALGEBRA FINAL EXAM
QUESTIONS WITH COMPLETE
ANSWERS
linear equation - Answer-in variables x1,x2,..,xn. An equation that can be put into the
form a1x1+a2x2+...+anxn=b , ai,b∈R
particular solution to a linear equation - Answer-an assignment of values to the
variables that satisfy a1x1+a2x2+...+anxn=b , ai,b∈R
general solution to a linear equation - Answer-the set of all solutions to a1x1+a2x2+...
+anxn=b , ai,b∈R
mn linear system of equations - Answer-any system of m equations in n variables.
x1,x2,...,xn that can be put into the form...
a11x1+...+a1nxn=b1
a21x1+...+a2nxn=b2 ai,b∈R
.....................................
an1x1+...+amnxn=bn
particular solution a linear system of equations - Answer-an assignment of the values to
the variables x1,x2,...,xn such that all the equations are satisfied simultaneously
general solution to a linear system of equations - Answer-the set of all possible
solutions to...
a11x1+...+a1nxn=b1
a21x1+...+a2nxn=b2 ai,b∈R
.....................................
an1x1+...+amnxn=bn
consistent linear system of equations - Answer-an mn linear system of equations that
has either one solution or infinitely many that satisfy all the equations simultaneously.
augmented matrix of a linear system of equations - Answer-a matrix of numbers in
which each row represents the constants a1,a2,...,an from one equation and each
column represents one variable x1,x2,...,xn
three types of row operations - Answer-1. Exchange two rows
2. Multiple a row by a nonzero, real number scalar
3. Multiply a row by a nonzero scalar and add it to another row
row echelon form - Answer-i. rows consisting entirely of zeros are grouped together at
the bottom of the matrix
, ii. in any 2 successive rows that do not consist entirely of zeros, the leading (leftmost
nonzero entry) in the lower row is farther to the right than the leading entry in the higher
row.
pivot entry - Answer-the leftmost nonzero entry in row i
pivot positions - Answer-the ij position of the pivot entry in row i
pivot columns - Answer-A column that contains a pivot entry
reduced row echelon form - Answer-i. the pivot entry in every row (if it has one) is 1
ii. all the entries in the pivot column, except the pivot entry, must be zero.
free variable - Answer-a variable that is not associated with a pivot column
matrix equality - Answer-two matrices are equal if they have the same dimensions and
their corresponding entries are the same
matrix addition/subtraction - Answer-To add matrices A and B with the same
dimensions, add corresponding elements Similarly, to subtract matrices A and B with
the same dimensions, subtract corresponding elements.
matrix-scalar multiplication - Answer-component wise multiplication by the scalar
matrix-matrix multiplication - Answer-[mxn]x[nxp]=[mxp] Cij =∑ aikbkj ∀i,j
identity matrix In - Answer-a square matrix whose only nonzero entries are on the main
diagonal, all nonzero diagonal entries are 1
matrix transpose - Answer-Cij=aji, where C denotes the transposed matrix
(switch columns and row in A)
matrix trace - Answer-tr(A)=a11+a22+...+ann (add diagonals)
matrix inverse - Answer-A^-1 st. AA^1=A^-1A=In
invertible matrix - Answer-A is an invertible matrix if...
ii. Ax=0 has a unique solution
iii. rrefA=In
iv. A is expressible as the product of elementary matrices
v. Ax=b is consistent for every choice of b
vi. Ax=b has a unique solution for every choice b
matrix power A^n - Answer-[A][A]....[A] n-times
QUESTIONS WITH COMPLETE
ANSWERS
linear equation - Answer-in variables x1,x2,..,xn. An equation that can be put into the
form a1x1+a2x2+...+anxn=b , ai,b∈R
particular solution to a linear equation - Answer-an assignment of values to the
variables that satisfy a1x1+a2x2+...+anxn=b , ai,b∈R
general solution to a linear equation - Answer-the set of all solutions to a1x1+a2x2+...
+anxn=b , ai,b∈R
mn linear system of equations - Answer-any system of m equations in n variables.
x1,x2,...,xn that can be put into the form...
a11x1+...+a1nxn=b1
a21x1+...+a2nxn=b2 ai,b∈R
.....................................
an1x1+...+amnxn=bn
particular solution a linear system of equations - Answer-an assignment of the values to
the variables x1,x2,...,xn such that all the equations are satisfied simultaneously
general solution to a linear system of equations - Answer-the set of all possible
solutions to...
a11x1+...+a1nxn=b1
a21x1+...+a2nxn=b2 ai,b∈R
.....................................
an1x1+...+amnxn=bn
consistent linear system of equations - Answer-an mn linear system of equations that
has either one solution or infinitely many that satisfy all the equations simultaneously.
augmented matrix of a linear system of equations - Answer-a matrix of numbers in
which each row represents the constants a1,a2,...,an from one equation and each
column represents one variable x1,x2,...,xn
three types of row operations - Answer-1. Exchange two rows
2. Multiple a row by a nonzero, real number scalar
3. Multiply a row by a nonzero scalar and add it to another row
row echelon form - Answer-i. rows consisting entirely of zeros are grouped together at
the bottom of the matrix
, ii. in any 2 successive rows that do not consist entirely of zeros, the leading (leftmost
nonzero entry) in the lower row is farther to the right than the leading entry in the higher
row.
pivot entry - Answer-the leftmost nonzero entry in row i
pivot positions - Answer-the ij position of the pivot entry in row i
pivot columns - Answer-A column that contains a pivot entry
reduced row echelon form - Answer-i. the pivot entry in every row (if it has one) is 1
ii. all the entries in the pivot column, except the pivot entry, must be zero.
free variable - Answer-a variable that is not associated with a pivot column
matrix equality - Answer-two matrices are equal if they have the same dimensions and
their corresponding entries are the same
matrix addition/subtraction - Answer-To add matrices A and B with the same
dimensions, add corresponding elements Similarly, to subtract matrices A and B with
the same dimensions, subtract corresponding elements.
matrix-scalar multiplication - Answer-component wise multiplication by the scalar
matrix-matrix multiplication - Answer-[mxn]x[nxp]=[mxp] Cij =∑ aikbkj ∀i,j
identity matrix In - Answer-a square matrix whose only nonzero entries are on the main
diagonal, all nonzero diagonal entries are 1
matrix transpose - Answer-Cij=aji, where C denotes the transposed matrix
(switch columns and row in A)
matrix trace - Answer-tr(A)=a11+a22+...+ann (add diagonals)
matrix inverse - Answer-A^-1 st. AA^1=A^-1A=In
invertible matrix - Answer-A is an invertible matrix if...
ii. Ax=0 has a unique solution
iii. rrefA=In
iv. A is expressible as the product of elementary matrices
v. Ax=b is consistent for every choice of b
vi. Ax=b has a unique solution for every choice b
matrix power A^n - Answer-[A][A]....[A] n-times
