LINEAR ALGEBRA EXAM QUESTIONS
AND ANSWERS
linear equation - Answer-An equation that can be written as a1x1 + a2x2 + ... = b; a1,
a2, etc. are real or complex numbers known in advance
consistent system - Answer-Has one or infinitely many solutions
inconsistent system - Answer-Has no solution
leading entry - Answer-Leftmost non-zero entry in a non-zero row
Echelon form - Answer-1. All nonzero rows are above any all zero rows; 2. Each leading
entry is in a column to the right of the previous leading entry; 3. All entries below a
leading entry in its column are zeros
Reduced Echelon Form - Answer-Same as echelon form, except all leading entries are
1; each leading 1 is the only non-zero entry in its row; there is only one unique reduced
echelon form for every matrix
Span - Answer-the collection of all vectors in R^n that can be written as c1v1 + c2v2
+ ... (where c1, c2, etc. are constants)
Ax = b - Answer-1. For each b in R^n, Ax = b has a solution; 2. Each b is a linear
combination of A; 3. The columns of A span R^n; 4. A has a pivot position in each row
pivot position - Answer-A position in the original matrix that corresponds to a leading 1
in a reduced echelon matrix
pivot column - Answer-A column that contains a pivot position
homogeneous - Answer-A system that can be written as Ax = 0; the x = 0 solution is a
TRIVIAL solution
independent - Answer-If only the trivial solution exists for a linear equation; the columns
of A are independent if only the trivial solution exists
dependent - Answer-If non-zero weights that satisfy the equation exist; if there are more
vectors than there are entries
transformation - Answer-assigns each vector x in R^n a vector T(x) in R^m
Matrix multiplication warnings - Answer-1. AB != BA ; 2. If AB = AC, B does not
necessarily equal C; 3. If AB = 0, it cannot be concluded that either A or B is equal to 0
AND ANSWERS
linear equation - Answer-An equation that can be written as a1x1 + a2x2 + ... = b; a1,
a2, etc. are real or complex numbers known in advance
consistent system - Answer-Has one or infinitely many solutions
inconsistent system - Answer-Has no solution
leading entry - Answer-Leftmost non-zero entry in a non-zero row
Echelon form - Answer-1. All nonzero rows are above any all zero rows; 2. Each leading
entry is in a column to the right of the previous leading entry; 3. All entries below a
leading entry in its column are zeros
Reduced Echelon Form - Answer-Same as echelon form, except all leading entries are
1; each leading 1 is the only non-zero entry in its row; there is only one unique reduced
echelon form for every matrix
Span - Answer-the collection of all vectors in R^n that can be written as c1v1 + c2v2
+ ... (where c1, c2, etc. are constants)
Ax = b - Answer-1. For each b in R^n, Ax = b has a solution; 2. Each b is a linear
combination of A; 3. The columns of A span R^n; 4. A has a pivot position in each row
pivot position - Answer-A position in the original matrix that corresponds to a leading 1
in a reduced echelon matrix
pivot column - Answer-A column that contains a pivot position
homogeneous - Answer-A system that can be written as Ax = 0; the x = 0 solution is a
TRIVIAL solution
independent - Answer-If only the trivial solution exists for a linear equation; the columns
of A are independent if only the trivial solution exists
dependent - Answer-If non-zero weights that satisfy the equation exist; if there are more
vectors than there are entries
transformation - Answer-assigns each vector x in R^n a vector T(x) in R^m
Matrix multiplication warnings - Answer-1. AB != BA ; 2. If AB = AC, B does not
necessarily equal C; 3. If AB = 0, it cannot be concluded that either A or B is equal to 0
