where e1,e2 are the standard basis vectors for ℝ2. Thus
e1 =
(1,0)
,e2 =
(0,1)
. Now suppose
( )
1 1
E = 0 1
where E is the elementary matrix which takes the third row and adds to the first. Draw
{Ex : x ∈ P (e1,e2)}.
In other words, draw the result of doing E to the vectors in P
(e1,e2)
. Next draw the results of doing
the other elementary matrices to P
(e1,e2)
. An elementary matrix is one which is obtained from
doing one of the row operations to the identity matrix.
Determine which matrices are in row reduced echelon form.
(a)
( )
1 2 0
0 1 7
(b)
( )
1 0 0 0
|( 0 0 1 2 |)
0 0 0 0
(c)
( )
1 1 0 0 0 5
|( 0 0 1 2 0 4 |)
0 0 0 0 1 3
Row reduce the following matrices to obtain the row reduced echelon form. List the pivot columns in
the original matrix.
(a)
( )
| 1 2 0 3 |
( 2 1 2 2 )
1 1 0 3
(b)
( )
| 1 2 3 |
|| 2 1 − 2 ||
( 3 0 0 )
3 2 1
(c)
( )
1 2 1 3
|( − 3 2 1 0 |)
3 2 1 1
Find the rank of the following matrices. If the rank is r, identify r columns in the original matrixwhich have the property that every other column may be written as a linear combination of these.
Also find a basis for column space of the matrices.
{u = (u1,u2,u3,u4) ∈ ℝ4 : ui ≥ 0 for each i = 1,2,3,4}
. Is M a subspace?
Explain.
Let w,w1 be given vectors in ℝ4 and define
{ 4 }
M = u = (u1,u2,u3,u4) ∈ ℝ : w ⋅u = 0 and w1 ⋅u = 0 .
Is M a subspace? Explain.
Let M =
{u = (u1,u2,u3,u4) ∈ ℝ4 : |u1| ≤ 4}
. Is M a subspace? Explain.
Let M =
{ }
u = (u1,u2,u3,u4) ∈ ℝ4 : sin (u1) = 1
. Is M a subspace? Explain.
Suppose
{x1,⋅⋅⋅,xk}
is a set of vectors from Fn. Show that span
(x1,⋅⋅⋅,xk)
contains
0.
Prove the following theorem: If A,B are n×n matrices and if AB = I, then BA = I and B = A−1.
Hint:First note that if AB = I, then it must be the case that A is onto. Explain why
this requires span
(columns of A)
= Fn. Now explain why, this requires A to be one to
one. Next explain why A
(BA − I)
= 0 and why the fact that A is one to one implies
BA = I.
Here are three vectors. Determine whether they are linearly independent or linearly dependent.
( )T ( )T ( )T
1 2 0 , 2 0 1 , 3 0 0
Here are three vectors. Determine whether they are linearly independent or linearly dependent.
( )T ( )T ( )T
4 2 0 , 2 2 1 , 0 2 2
Here are three vectors. Determine whether they are linearly independent or linearly dependent.
( )T ( )T ( )T
1 2 3 , 4 5 1 , 3 1 0
Here are four vectors. Determine whether they span ℝ3. Are these vectors linearly independent?
( ) ( ) ( ) ( )
1 2 3 T , 4 3 3 T , 3 1 0 T , 2 4 6 T
Here are four vectors. Determine whether they span ℝ3. Are these vectors linearly independent?
( ) ( ) ( ) ( )
1 2 3 T , 4 3 3 T , 3 2 0 T , 2 4 6 T
Determine whether the following vectors are a basis for ℝ3. If they are, explain why they are and if
they are not, give a reason and tell whether they span ℝ3.
( )T ( )T ( )T ( )T
1 0 3 , 4 3 3 , 1 2 0 , 2 4 0
Determine whether the following vectors are a basis for ℝ3. If they are, explain why they are and if
they are not, give a reason and tell whether they span ℝ3.
( )T ( )T ( )T
1 0 3 , 0 1 0 , 1 2 0
Determine whether the following vectors are a basis for ℝ3. If they are, explain why they are and if
they are not, give a reason and tell whether they span ℝ3.
( ) ( ) ( ) ( )
1 0 3 T , 0 1 0 T , 1 2 0 T , 0 0 0 T
Determine whether the following vectors are a basis for ℝ3. If they are, explain why they are and if
they are not, give a reason and tell whether they span ℝ3.
( ) ( ) ( ) ( )
1 0 3 T , 0 1 0 T , 1 1 3 T , 0 0 0 T
Is this set of vectors a subspace of ℝ3? If so, explain why, give a basis for the subspace and find its
dimension.
Consider the vectors of the form
(( ) )
|||{| 2t+ 3s+ u | |||}
|| s− t || : s,t,u ∈ ℝ .
|||( t+ s ) |||
( u )
Is this set of vectors a subspace of ℝ4? If so, explain why, give a basis for the subspace and find its
dimension.
Consider the vectors of the form
(|( 2t+ u ) )|
||{| t+ 3u | ||}
|| || : s,t,u,v ∈ ℝ .
|||(( t+ s+ v ) |||)
u
Is this set of vectors a subspace of ℝ4? If so, explain why, give a basis for the subspace and find its
dimension.
If you have 5 vectors in F5 and the vectors are linearly independent, can it always be concluded they
span F5? Explain.
If you have 6 vectors in F5, is it possible they are linearly independent? Explain.
Suppose A is an m × n matrix and
{w1,⋅⋅⋅,wk}
is a linearly independent set of vectors in
A
n
(F )
⊆ Fm. Now suppose A
(zi)
= wi. Show
{z1,⋅⋅⋅,zk}
is also independent.
Suppose V,W are subspaces of Fn. Show V ∩W defined to be all vectors which are in both V and W
is a subspace also.
Suppose V and W both have dimension equal to 7 and they are subspaces of F10. What are the
possibilities for the dimension of V ∩ W? Hint: Remember that a linear independent set can be
extended to form a basis.
Suppose V has dimension p and W has dimension q and they are each contained in a subspace, U
which has dimension equal to n where n > max
(p,q)
. What are the possibilities for the dimension
of V ∩ W? Hint: Remember that a linear independent set can be extended to form a
basis.
If b≠0, can the solution set of Ax = b be a plane through the origin? Explain.
Suppose a system of equations has fewer equations than variables and you have found a solution to
this system of equations. Is it possible that your solution is the only one? Explain.
Suppose a system of linear equations has a 2 × 4 augmented matrix and the last column is a pivot
column. Could the system of linear equations be consistent? Explain.
Suppose the coefficient matrix of a system of n equations with n variables has the property that every
column is a pivot column. Does it follow that the system of equations must have a solution? If so,
must the solution be unique? Explain.
Suppose there is a unique solution to a system of linear equations. What must be true of the pivot
columns in the augmented matrix.
State whether each of the following sets of data are possible for the matrix equation Ax = b. If
possible, describe the solution set. That is, tell whether there exists a unique solution no solution or
infinitely many solutions.
A is a 5 × 6 matrix, rank
(A)
= 4 and rank
(A|b)
= 4. Hint:This says b is in the span
of four of the columns. Thus the columns are not independent.
A is a 3 × 4 matrix, rank
(A)
= 3 and rank
(A |b )
= 2.
A is a 4 × 2 matrix, rank
(A)
= 4 and rank
(A|b)
= 4. Hint:This says b is in the span
of the columns and the columns must be independent.
A is a 5 × 5 matrix, rank
(A)
= 4 and rank
(A|b)
= 5. Hint:This says b is not in the
span of the columns.
A is a 4 × 2 matrix, rank
(A)
= 2 and rank
(A |b )
= 2.
Suppose A is an m×n matrix in which m ≤ n. Suppose also that the rank of A equals m. Show that
A maps Fn onto Fm. Hint: The vectors e1,
⋅⋅⋅
,em occur as columns in the row reduced echelon form
for A.
Suppose A is an m×n matrix in which m ≥ n. Suppose also that the rank of A equals n. Show that
A is one to one. Hint: If not, there exists a vector x such that Ax = 0, and this implies at least one
column of A is a linear combination of the others. Show this would require the column rank to be less
than n.
Explain why an n × n matrix A is both one to one and onto if and only if its rank is
n.
Suppose A is an m × n matrix and B is an n × p matrix. Show that
dim (ker(AB )) ≤ dim (ker(A))+ dim(ker(B)).
Hint:Consider the subspace, B
(Fp)
∩ ker
(A )
and suppose a basis for this subspace
is
{w1,⋅⋅⋅,wk }.
Now suppose
{u1,⋅⋅⋅,ur}
is a basis for ker
(B)
. Let
{z1,⋅⋅⋅,zk}
be such that Bzi = wi and argue
that
ker(AB ) ⊆ span(u1,⋅⋅⋅,ur,z1,⋅⋅⋅,zk).
Here is how you do this. Suppose ABx = 0. Then Bx ∈ ker
(A)
∩ B
p
(F )
and so Bx = ∑i=1kBzi
showing that
∑k
x − zi ∈ ker(B ).
i=1
Explain why Ax = 0 always has a solution even when A−1 does not exist.
What can you conclude about A if the solution is unique?
What can you conclude about A if the solution is not unique?
Let A be an n×n matrix and let x be a nonzero vector such that Ax = λx for some scalar λ. When
this occurs, the vector x is called an eigenvector and the scalar λ is called an eigenvalue. It turns
out that not every number is an eigenvalue. Only certain ones are. Why? Hint:Show that if
Ax = λx, then
(A − λI)
x = 0. Explain why this shows that
(A − λI)
is not one to one and not
onto.
Let A be an n × n matrix and consider the matrices
{ 2 n2}
I,A, A ,⋅⋅⋅,A
. Explain why there exist
scalars, ci not all zero such that
n2
∑ ciAi = 0.
i=1
Then argue there exists a polynomial, p
(λ)
of the form
m m −1
λ + dm−1λ + ⋅⋅⋅+ d1λ + d0
such that p
(A )
= 0 and if q
(λ)
is another polynomial such that q
(A)
= 0, then q
(λ)
is of the form
p
(λ)
l
(λ)
for some polynomial, l
(λ)
. This extra special polynomial, p
(λ)
is called the minimalpolynomial. Hint:You might consider an n×n matrix as a vector in Fn2. What would be a basis
for this set of matrices?