A First Course in Linear Algebra, 2017a

9.7. Isomorphisms 509
Exercises
Exercise 9.7.1 Let V and W be subspaces of R n and R m respectively and let T : V → W be a linear
transformation. Suppose that {T⃗v 1 ,···,T⃗v r } is linearly independent. Show that it must be the case that
{⃗v 1 ,···,⃗v r } is also linearly independent.
Exercise 9.7.2 Let
Let T⃗x = A⃗x where A is the matrix
Give a basis for im(T ).
Exercise 9.7.3 Let
Let T⃗x = A⃗x where A is the matrix
⎧⎡
⎪⎨
V = span ⎢
⎣
⎪⎩
⎡
⎢
⎣
⎧⎡
⎪⎨
V = span ⎢
⎣
⎪⎩
⎡
⎢
⎣
1
1
2
0
⎤
⎡
⎥
⎦ , ⎢
⎣
0
1
1
1
1 1 1 1
0 1 1 0
0 1 2 1
1 1 1 2
1
0
0
1
⎤
⎡
⎥ ⎦ , ⎢
⎣
1
1
1
1
1 1 1 1
0 1 1 0
0 1 2 1
1 1 1 2
⎤
⎡
⎥
⎦ , ⎢
⎤
⎥
⎦
⎣
⎤ ⎡
⎥
⎦ , ⎢
⎣
⎤
⎥
⎦
1
1
0
1
1
4
4
1
⎤⎫
⎪⎬
⎥
⎦
⎪⎭
⎤⎫
⎪⎬
⎥
⎦
⎪⎭
Find a basis for im(T ). In this case, the original vectors do not form an independent set.
Exercise 9.7.4 If {⃗v 1 ,···,⃗v r } is linearly independent and T is a one to one linear transformation, show
that {T⃗v 1 ,···,T⃗v r } is also linearly independent. Give an example which shows that if T is only linear, it
can happen that, although {⃗v 1 ,···,⃗v r } is linearly independent, {T⃗v 1 ,···,T⃗v r } is not. In fact, show that it
can happen that each of the T⃗v j equals 0.
Exercise 9.7.5 Let V and W be subspaces of R n and R m respectively and let T : V → W be a linear
transformation. Show that if T is onto W and if {⃗v 1 ,···,⃗v r } is a basis for V, then span{T⃗v 1 ,···,T⃗v r } =
W.
Exercise 9.7.6 Define T : R 4 → R 3 as follows.
⎡
T⃗x = ⎣
Find a basis for im(T ). Also find a basis for ker(T ).
3 2 1 8
2 2 −2 6
1 1 −1 3
⎤
⎦⃗x