Linear Algebra

Section III. Computing Linear Maps 209
� 2.13 Decide if 1 + 2x is in the range of the map from R 3 to P2 represented with
respect to E3 and 〈1, 1+x 2 ,x〉 by this matrix.
� �
1 3 0
0 1 0
1 0 1
2.14 Example 2.8 gives a matrix that is nonsingular, and is therefore associated
with maps that are nonsingular.
(a) Find the set of column vectors representing the members of the nullspace of
any map represented by this matrix.
(b) Find the nullity of any such map.
(c) Find the set of column vectors representing the members of the rangespace
of any map represented by this matrix.
(d) Find the rank of any such map.
(e) Check that rank plus nullity equals the dimension of the domain.
� 2.15 Because the rank of a matrix equals the rank of any map it represents, if
one matrix represents two different maps H =Rep B,D(h) =Rep ˆ B, ˆ D ( ˆ h)(where
h, ˆ h: V → W ) then the dimension of the rangespace of h equals the dimension of
the rangespace of ˆ h. Must these equal-dimensioned rangespaces actually be the
same?
� 2.16 Let V be an n-dimensional space with bases B and D. Consider a map that
sends, for �v ∈ V , the column vector representing �v with respect to B to the column
vector representing �v with respect to D. Show that is a linear transformation of
R n .
2.17 Example 2.2 shows that changing the pair of bases can change the map that
a matrix represents, even though the domain and codomain remain the same.
Could the map ever not change? Is there a matrix H, vectorspacesV and W ,
and associated pairs of bases B1,D1 and B2,D2 (with B1 �= B2 or D1 �= D2 or
both) such that the map represented by H with respect to B1,D1 equals the map
represented by H with respect to B2,D2?
� 2.18 A square matrix is a diagonal matrix if it is all zeroes except possibly for the
entries on its upper-left to lower-right diagonal—its 1, 1entry,its2, 2entry,etc.
Show that a linear map is an isomorphism if there are bases such that, with respect
to those bases, the map is represented by a diagonal matrix with no zeroes on the
diagonal.
2.19 Describe geometrically the action on R 2 of the map represented with respect
to the standard bases E2, E2 by this matrix.
� �
3 0
0 2
Do the same for these. �1 �
0
�
0
�
1
�
1
�
3
0 0 1 0 0 1
2.20 The fact that for any linear map the rank plus the nullity equals the dimension
of the domain shows that a necessary condition for the existence of a homomorphism
between two spaces, onto the second space, is that there be no gain in
dimension. That is, where h: V → W is onto, the dimension of W must be less
than or equal to the dimension of V .
(a) Show that this (strong) converse holds: no gain in dimension implies that