S OCIAL E COLOGY W EEK 6: I NVERTIBLE LINEAR TRANSFORMATIONS
M ATH 121A W INTER 2017
1. Let T : V ! W be a linear transformation such that N (T ) = {0}. Prove that T is one-to-one.
(Must it be true that V and W have the same finite dimension?)
2. Recall that a linear transformation T : V ! W is invertible if and only if it is both one-to-one
and onto.
a. Let V, W denote finite-dimensional vector spaces. Using the Rank-Nullity theorem, prove
that if dim V < dim W , then T is not onto.
b. Let V, W denote finite-dimensional vector spaces. Using the Rank-Nullity theorem, prove
that if dim V > dim W , then T is not one-to-one.
c. Let V, W denote finite-dimensional vector spaces. Prove that if V, W have different dimensions, then T is not an isomorphism.
3. Prove directly from the definition that an isomorphism is one-to-one and onto.
4. Let V = R2 and consider the ordered basis
{(2, 1), (3, 1)}.
= {(1, 0), (0, 1)} and the ordered basis
0
=
0
a. Compute the change of coordinate matrix [IV ] .
b. Compute the change of coordinate matrix [IV ] 0 .
c. What happens when you multiply these two matrices together, in either order? Does that
make sense?
5. Give an explicit isomorphism from V = {(x, y, z) | x + y 2z = 0} to W = R2 and its inverse.
Write the corresponding matrices and check that they multiply to give the identity matrix.
6. Let T : V ! W and U : W ! Z be linear transformations. Assume U T is invertible and T is
invertible. Prove that U is also invertible.
7. a. Let V be a finite dimensional vector space, and assume T : V ! V is a linear transformation.
Prove that T is onto if and only if T is one-to-one.
b. Let V, W be finite dimensional vector spaces, and assume T : V ! W and U : W ! V are
linear transformations satisfying U T = IV . Show that it’s not necessarily the case that T is
invertible.
8. Let V, W be finite-dimensional vector spaces over a field F . There is only one linear transformation for which the corresponding matrix does not depend on the choice of ordered basis for
V and ordered basis for W . What linear transformation is that?
9. Prove that if T : V ! W is invertible, then its inverse is unique. (Hint. Assume U1 , U2 are two
inverses. Consider U2 T U1 to prove that U1 = U2 .)
10. Only a square matrix can be invertible. What is the corresponding statement for linear transformations?