LINEAR ALGEBRA CHAPTER 1 EXAM
QUESTIONS AND ANSWERS
Linear Equation - Answer-a1x1+a2x2+...+anxn=b
b and coefficients are real numbers
System of Linear Equations - Answer-collection of one or more linear equations with the
same variables
Solution of System - Answer-Ordered list of numbers which make each of the equations
true when plugged in
Linear Systems - Answer-can be:
-no sol'n (inconsistent)
-1 sol'n (consistent, unique)
-infinite sol'n (consistent)
Consistent - Answer-System with at least one solution
Inconsistent - Answer-System with no solutions
Unique - Answer-Exactly 1 solution exists
Coefficient matrix - Answer-Coefficients ONLY
Augmented Matrix - Answer-Coefficients and answer (b column)
Size of Matrix - Answer-m x n or rows x col
Elementary Row Operations - Answer-Replacement (addition) replace one row by sum
of itself and different row
Interchange- swap the positions of 2 rows
Scaling- multiply all entries in a row by a nonzero constant
Same Solution Set - Answer-Augmented Matrices of two linear systems are row
equivalent
Echelon Form - Answer-- all nonzero rows (at least one nonzero entry in it) are above
all zero rows
- each leading entry (leftmost nonzero entry) of a row is to the right of the leading entry
above it
Reduced Echelon Form - Answer-- in echelon form
- each leading entry is a one
, -each leading one is the only nonzero entry in its column
Thm 1 - Answer-Every matrix is row equivalent to one and only one reduced echelon
matrix
Every matrix can be row reduced to a reduced echelon matrix
Pivot Position - Answer-is a location in matrix A which corresponds to a leading 1 in the
reduced echelon form of A
Pivot Column - Answer-column with a pivot position
Basic Variables - Answer-x1, x2
Non-basic variables - Answer-Free variables
Parametric Description - Answer-Any choice of free variable gives a different solution
and any solution is determined by choice of free variable (Infinite Solutions)
Thm 2 (Existence and Uniqueness thm) - Answer--Linear system is consistent IFF the
rightmost column of the augmented matrix is NOT a pivot column, that is IFF an echelon
form of the augmented matrix has no row of the form:
[0 .... 0 b] with b nonzero
-Homogenous system has nontrivial solution IFF free variable
Consistent Linear System - Answer-Solution set:
-Unique Solution, when there are no free variables
-Infinitely Many Solutions, where there are free variables
-every column except rightmost has a pivot
Vector - Answer--Matrix with one column
-set of all vectors with n rows R^n
-think of any mxn matrix as ordered list of n vectors in R^m
Algebraic Properties of R^n - Answer-I) u+v = v+u
II) (u+v)+w = u+(v+w)
III) u+0=0+u
IV)u+(-u)=0
V) c(u+v)=cu+cv
VI)(c+d)u=cu+du
VII) c(du)=(cd)u
Linear Combination - Answer-V1, V2, ..., VP are in R^n and c1, c2.., cP are in R then
c1v1, c2v2+...+cpvp
coefficients are c
QUESTIONS AND ANSWERS
Linear Equation - Answer-a1x1+a2x2+...+anxn=b
b and coefficients are real numbers
System of Linear Equations - Answer-collection of one or more linear equations with the
same variables
Solution of System - Answer-Ordered list of numbers which make each of the equations
true when plugged in
Linear Systems - Answer-can be:
-no sol'n (inconsistent)
-1 sol'n (consistent, unique)
-infinite sol'n (consistent)
Consistent - Answer-System with at least one solution
Inconsistent - Answer-System with no solutions
Unique - Answer-Exactly 1 solution exists
Coefficient matrix - Answer-Coefficients ONLY
Augmented Matrix - Answer-Coefficients and answer (b column)
Size of Matrix - Answer-m x n or rows x col
Elementary Row Operations - Answer-Replacement (addition) replace one row by sum
of itself and different row
Interchange- swap the positions of 2 rows
Scaling- multiply all entries in a row by a nonzero constant
Same Solution Set - Answer-Augmented Matrices of two linear systems are row
equivalent
Echelon Form - Answer-- all nonzero rows (at least one nonzero entry in it) are above
all zero rows
- each leading entry (leftmost nonzero entry) of a row is to the right of the leading entry
above it
Reduced Echelon Form - Answer-- in echelon form
- each leading entry is a one
, -each leading one is the only nonzero entry in its column
Thm 1 - Answer-Every matrix is row equivalent to one and only one reduced echelon
matrix
Every matrix can be row reduced to a reduced echelon matrix
Pivot Position - Answer-is a location in matrix A which corresponds to a leading 1 in the
reduced echelon form of A
Pivot Column - Answer-column with a pivot position
Basic Variables - Answer-x1, x2
Non-basic variables - Answer-Free variables
Parametric Description - Answer-Any choice of free variable gives a different solution
and any solution is determined by choice of free variable (Infinite Solutions)
Thm 2 (Existence and Uniqueness thm) - Answer--Linear system is consistent IFF the
rightmost column of the augmented matrix is NOT a pivot column, that is IFF an echelon
form of the augmented matrix has no row of the form:
[0 .... 0 b] with b nonzero
-Homogenous system has nontrivial solution IFF free variable
Consistent Linear System - Answer-Solution set:
-Unique Solution, when there are no free variables
-Infinitely Many Solutions, where there are free variables
-every column except rightmost has a pivot
Vector - Answer--Matrix with one column
-set of all vectors with n rows R^n
-think of any mxn matrix as ordered list of n vectors in R^m
Algebraic Properties of R^n - Answer-I) u+v = v+u
II) (u+v)+w = u+(v+w)
III) u+0=0+u
IV)u+(-u)=0
V) c(u+v)=cu+cv
VI)(c+d)u=cu+du
VII) c(du)=(cd)u
Linear Combination - Answer-V1, V2, ..., VP are in R^n and c1, c2.., cP are in R then
c1v1, c2v2+...+cpvp
coefficients are c
