LINEAR ALGEBRA EXAM 1 (CH. 1)
QUESTIONS WITH COMPLETE
SOLUTIONS
linear equation in the variables x1,x2,x3,...xn - Answer-an equation that can be written
as a1x1+a2x2+a3x3+...+anxn=b, where the a's are constants and not all 0
homogenous linear equation - Answer-of the form a1x1+a2x2+a3x3+...+anxn=0
linear system (or system of linear equations) - Answer-a finite set of linear equations
unknowns - Answer-variables used in a linear system
solution of a linear system - Answer-a sequence of values for the variables
s1,s2,s3,...sn that make all of the equations in the system true
a system of equations is inconsistent if - Answer-it has no solutions
a system of equations is consistent if - Answer-it has at least one solution
a system of 3 linear equations in 2 variables has 3 cases: - Answer-one solution, no
solution, infinitely many solutions
in a system of 3 linear equations in 2 variables, what does one solution mean - Answer-
the solution is the point of intersection of the 2 lines
in a system of 3 linear equations in 2 variables, what does no solution mean - Answer-
the lines are parallel
in a system of 3 linear equations in 2 variables, what does infinitely many solutions
mean - Answer-every point on the line is a solution, the lines are coincident (same line)
(on top of each other)
for n=2, the graphs of linear equations are - Answer-lines
for n=3, the graphs of linear equations are - Answer-planes in 3D
what does no solution mean for n=3 (planes) - Answer-no common intersection among
the planes
what does one solution mean for n=3 (planes) - Answer-there is one common
intersection among the planes
, what does infinitely many solution mean for n=3 (planes) - Answer-all three planes
intersect along the same line
use __________ ___________ to state the solution set of a system with infinitely many
solutions - Answer-parametric equations
general solution - Answer-a set of parametric equations that generate all solutions to a
system of equations
matrix - Answer-a rectangular array of numbers, written inside brackets
elements/entries of a matrix - Answer-the numbers in a matrix
coefficient matrix - Answer-a matrix that contains only the coefficients of a system of
equations
augmented matrix - Answer-coefficient matrix with the added column of solutions (add a
vertical line)
3 elementary row operations - Answer-1. multiply each element in a row by a nonzero
constant
2. interchange 2 rows
3. add a nonzero multiple of a row to another row
a matrix is in row echelon form (ref) if - Answer-1. each row that contains a nonzero
element has 1 as its first nonzero element (leading 1's)
2. all rows consisting of only zeros are together at the bottom of the matrix
3. for any two rows with leading 1's, the leading 1 of the lower row is to the right of the
leading one the higher row
**below each leading 1, there are only zeros**
a matrix is in reduced row echelon form (rref) if - Answer-1. it is in row echelon form
2. any column containing a leading 1 has zeros for all other entries
**below and above each leading 1 are only zeros**
to solve a system from its ref, turn it into a system of equations, and start at the
_______ and work your way ______ - Answer-start at bottom row and work up
leading variables - Answer-variables corresponding to the pivot columns (columns
containing leading 1s)
free variables - Answer-variables corresponding to non-pivot columns
QUESTIONS WITH COMPLETE
SOLUTIONS
linear equation in the variables x1,x2,x3,...xn - Answer-an equation that can be written
as a1x1+a2x2+a3x3+...+anxn=b, where the a's are constants and not all 0
homogenous linear equation - Answer-of the form a1x1+a2x2+a3x3+...+anxn=0
linear system (or system of linear equations) - Answer-a finite set of linear equations
unknowns - Answer-variables used in a linear system
solution of a linear system - Answer-a sequence of values for the variables
s1,s2,s3,...sn that make all of the equations in the system true
a system of equations is inconsistent if - Answer-it has no solutions
a system of equations is consistent if - Answer-it has at least one solution
a system of 3 linear equations in 2 variables has 3 cases: - Answer-one solution, no
solution, infinitely many solutions
in a system of 3 linear equations in 2 variables, what does one solution mean - Answer-
the solution is the point of intersection of the 2 lines
in a system of 3 linear equations in 2 variables, what does no solution mean - Answer-
the lines are parallel
in a system of 3 linear equations in 2 variables, what does infinitely many solutions
mean - Answer-every point on the line is a solution, the lines are coincident (same line)
(on top of each other)
for n=2, the graphs of linear equations are - Answer-lines
for n=3, the graphs of linear equations are - Answer-planes in 3D
what does no solution mean for n=3 (planes) - Answer-no common intersection among
the planes
what does one solution mean for n=3 (planes) - Answer-there is one common
intersection among the planes
, what does infinitely many solution mean for n=3 (planes) - Answer-all three planes
intersect along the same line
use __________ ___________ to state the solution set of a system with infinitely many
solutions - Answer-parametric equations
general solution - Answer-a set of parametric equations that generate all solutions to a
system of equations
matrix - Answer-a rectangular array of numbers, written inside brackets
elements/entries of a matrix - Answer-the numbers in a matrix
coefficient matrix - Answer-a matrix that contains only the coefficients of a system of
equations
augmented matrix - Answer-coefficient matrix with the added column of solutions (add a
vertical line)
3 elementary row operations - Answer-1. multiply each element in a row by a nonzero
constant
2. interchange 2 rows
3. add a nonzero multiple of a row to another row
a matrix is in row echelon form (ref) if - Answer-1. each row that contains a nonzero
element has 1 as its first nonzero element (leading 1's)
2. all rows consisting of only zeros are together at the bottom of the matrix
3. for any two rows with leading 1's, the leading 1 of the lower row is to the right of the
leading one the higher row
**below each leading 1, there are only zeros**
a matrix is in reduced row echelon form (rref) if - Answer-1. it is in row echelon form
2. any column containing a leading 1 has zeros for all other entries
**below and above each leading 1 are only zeros**
to solve a system from its ref, turn it into a system of equations, and start at the
_______ and work your way ______ - Answer-start at bottom row and work up
leading variables - Answer-variables corresponding to the pivot columns (columns
containing leading 1s)
free variables - Answer-variables corresponding to non-pivot columns
