LINEAR ALGEBRA EXAM 3: TRUE OR
FALSE- QUESTIONS AND ANSWERS
If {v1, . . . , vp−1} spans V , then S spans V - Answer-True
If S is linearly independent, then S is a basis for V - Answer-False
The set of all linear combinations of v1... vp is a vector space - Answer-True, This set is
Span{ , ... } 1 p v v , and every subspace is itself a vector space.
If {v1, . . . , vp−1} is a linearly independent set, then S is a linearly independent set -
Answer-False
If Span S= V, then some subset of S is a basis for V - Answer-True, by the spanning set
theorem
If dim V= P and Span S= V, then S cannot be linearly dependent - Answer-True, must
be linearly independent
The nonpivot columns of a matrix are always linearly dependent - Answer-False
Row operations on a matrix can change the linear dependence relations among the
rows of A - Answer-True, interchanging them can change dependence
Row operations on a matrix change the null space - Answer-False, does not change
ax=0
The rank of a matrix= the number of non zero rows - Answer-False
If B is obtained from a matrix A by several elementary row operations, then rank B=rank
A - Answer-True, row equivalent matrices have the same number of pivot columns
The nonzero rows of a matrix A form a basis for Row A - Answer-False, The nonzero
rows of A span Row A but they might not be linearly independent.
If matrices A and B have the same reduced echelon form, then Row A = Row B -
Answer-True, The nonzero rows of the reduced echelon form E form a basis for the row
space of each
matrix that is row equivalent to E.
If H is a subspace of R³ then there is a 3x3 matrix A such that H=Col A - Answer-True
R² is a two dimensional subspace of R³ - Answer-False, they're not related
FALSE- QUESTIONS AND ANSWERS
If {v1, . . . , vp−1} spans V , then S spans V - Answer-True
If S is linearly independent, then S is a basis for V - Answer-False
The set of all linear combinations of v1... vp is a vector space - Answer-True, This set is
Span{ , ... } 1 p v v , and every subspace is itself a vector space.
If {v1, . . . , vp−1} is a linearly independent set, then S is a linearly independent set -
Answer-False
If Span S= V, then some subset of S is a basis for V - Answer-True, by the spanning set
theorem
If dim V= P and Span S= V, then S cannot be linearly dependent - Answer-True, must
be linearly independent
The nonpivot columns of a matrix are always linearly dependent - Answer-False
Row operations on a matrix can change the linear dependence relations among the
rows of A - Answer-True, interchanging them can change dependence
Row operations on a matrix change the null space - Answer-False, does not change
ax=0
The rank of a matrix= the number of non zero rows - Answer-False
If B is obtained from a matrix A by several elementary row operations, then rank B=rank
A - Answer-True, row equivalent matrices have the same number of pivot columns
The nonzero rows of a matrix A form a basis for Row A - Answer-False, The nonzero
rows of A span Row A but they might not be linearly independent.
If matrices A and B have the same reduced echelon form, then Row A = Row B -
Answer-True, The nonzero rows of the reduced echelon form E form a basis for the row
space of each
matrix that is row equivalent to E.
If H is a subspace of R³ then there is a 3x3 matrix A such that H=Col A - Answer-True
R² is a two dimensional subspace of R³ - Answer-False, they're not related
