LINEAR ALGEBRA EXAM 2 REVIEW
GUIDE QUESTIONS WITH CORRECT
ANSWERS
Theorem: For an mxn matrix A, NulA is a subspace of _____. - Answer- R^n
A basis for a subspace H of Rn is ______. - Answer-a linearly independent set in H that
spans H
Theorem 13: The pivot columns of A form a basis for ____. - Answer- ColA
H = {0} is a subspace of Rn, commonly called the ____. - Answer-zero subspace
ColA is a subspace of ____ when A is mxn. - Answer-Rn
The ____ of an mxn matrix A is the set NulA of all solutions of the homogeneous
equation Ax=0 - Answer-null space
Fact: If a subspace H has a basis of p vectors, then every basis of H must consist of
___ vectors. - Answer-p
The DIMENSION of a nonzero subspace H is the number of vectors in _____. - Answer-
any basis of H.
The RANK of matrix A is _______. - Answer-the dimension of the column space of A.
(dim(ColA)).
In other words, the rank of A is ___ - Answer-the number of pivot columns of A.
Thm 14: Rank Theorem - Answer-If a matrix A has n columns, then rankA+dim(NulA) =
n.
Thm 15: Basis Theorem - Let H be a p-dimensional subspace of Rn. Any ____ is
automatically a basis for H. Also, any set of p vectors that ______ is automatically a
basis of H. - Answer-linearly independent set of exactly p vectors
spans H
Invertible Matrix Theorem (continued)
Let A be an nxn matrix.
The following are equivalent.
, a) A is invertible.
b) A is row equivalent to In.
c) A has n pivot positions.
d) Ax=0 has only the trivial solution.
e) The columns of A are linearly independent
f) the lin. trans. x->Ax is one to one.
g) Ax=b has a unique solution for any b in Rn.
h) The columns of A span Rn.
i) The lin. trans x->Ax maps Rn onto rn.
j) There exists an nxn matrix c such that CA=In.
k) There exists an nxn matrix D such that AD=In.
L) A^T is an invertible matrix.
m) ____________________
n)______________________
o)______________________
p)______________________
q)______________________
r)______________________
s) ______________________ - Answer-m) The columns of A form a basis for Rn
n) colA = Rn
o) dim(colA) = n
p) rankA = n
q)NulA = 0
r) dim(NulA) = 0
s) det(A) != 0
t) 0 is NOt an eigenvalue of A
Thm 2: If A is a triangular square matrix, then det(A) is _____. - Answer-the product of
the diagonal entries of A.
Thm 3 - Row Operations: Let A be a nxn matrix.
A) If a multiple of one row of matrix A is added to another to obtain matrix B, then det(A)
= ____.
B) If 2 rows of A are swapped to obtain B, then det(B) = ____.
C) If one row of A is multiplied by a scalar k to obtain B, then det(B) = _____. - Answer-
a) detB
b) -detA
c) k * det A
For any square matrix, det(a) = 0 iff any of the following are true:
1) A has _________.
2) columns of A are linearly _____.
GUIDE QUESTIONS WITH CORRECT
ANSWERS
Theorem: For an mxn matrix A, NulA is a subspace of _____. - Answer- R^n
A basis for a subspace H of Rn is ______. - Answer-a linearly independent set in H that
spans H
Theorem 13: The pivot columns of A form a basis for ____. - Answer- ColA
H = {0} is a subspace of Rn, commonly called the ____. - Answer-zero subspace
ColA is a subspace of ____ when A is mxn. - Answer-Rn
The ____ of an mxn matrix A is the set NulA of all solutions of the homogeneous
equation Ax=0 - Answer-null space
Fact: If a subspace H has a basis of p vectors, then every basis of H must consist of
___ vectors. - Answer-p
The DIMENSION of a nonzero subspace H is the number of vectors in _____. - Answer-
any basis of H.
The RANK of matrix A is _______. - Answer-the dimension of the column space of A.
(dim(ColA)).
In other words, the rank of A is ___ - Answer-the number of pivot columns of A.
Thm 14: Rank Theorem - Answer-If a matrix A has n columns, then rankA+dim(NulA) =
n.
Thm 15: Basis Theorem - Let H be a p-dimensional subspace of Rn. Any ____ is
automatically a basis for H. Also, any set of p vectors that ______ is automatically a
basis of H. - Answer-linearly independent set of exactly p vectors
spans H
Invertible Matrix Theorem (continued)
Let A be an nxn matrix.
The following are equivalent.
, a) A is invertible.
b) A is row equivalent to In.
c) A has n pivot positions.
d) Ax=0 has only the trivial solution.
e) The columns of A are linearly independent
f) the lin. trans. x->Ax is one to one.
g) Ax=b has a unique solution for any b in Rn.
h) The columns of A span Rn.
i) The lin. trans x->Ax maps Rn onto rn.
j) There exists an nxn matrix c such that CA=In.
k) There exists an nxn matrix D such that AD=In.
L) A^T is an invertible matrix.
m) ____________________
n)______________________
o)______________________
p)______________________
q)______________________
r)______________________
s) ______________________ - Answer-m) The columns of A form a basis for Rn
n) colA = Rn
o) dim(colA) = n
p) rankA = n
q)NulA = 0
r) dim(NulA) = 0
s) det(A) != 0
t) 0 is NOt an eigenvalue of A
Thm 2: If A is a triangular square matrix, then det(A) is _____. - Answer-the product of
the diagonal entries of A.
Thm 3 - Row Operations: Let A be a nxn matrix.
A) If a multiple of one row of matrix A is added to another to obtain matrix B, then det(A)
= ____.
B) If 2 rows of A are swapped to obtain B, then det(B) = ____.
C) If one row of A is multiplied by a scalar k to obtain B, then det(B) = _____. - Answer-
a) detB
b) -detA
c) k * det A
For any square matrix, det(a) = 0 iff any of the following are true:
1) A has _________.
2) columns of A are linearly _____.
