Homework 9
Homework 9
Due on Tuesday, April 7 (In class)
Exercise 1 (1 pt.)
Consider the transformation T : R4 → R4 given by
−1 1
0 −2
A=
1
1
2
2
the matrix
3 −3
0
4
.
−3 −1
−6 −2
In Exercise 1 of Homework 7 you found bases for the kernel and the image of T . Find the dimensions
of ker T and im T and explain how these fit in the Rank-Nullity Theorem.
[Hint: In case you did not solve Exercise 1 of Homework 7, or your solution is not available to you
right now, you can find the bases for ker T and im T on the solutions posted online.]
Exercise 2 (1 pt.)
Consider the subspace V of R3 with basis B = {~v1 , ~v2 }, where
3
1
~v1 = −2 and ~v2 = 0 .
1
0
(a) Write the vectors ~x1 = [ 2 2 1 ]t and ~x2 = [ 7 − 2 2 ]t in V in terms of the basis B, i.e. find
[~x1 ]B and [~x2 ]B (you do not need to verify that ~x1 and ~x2 are in V ).
(b) Find the vectors ~y1 , ~y2 in R3 given that [~y1 ]B = [ 3 − 1 ]t and [~y2 ]B = [ −4 0 ]t .
Exercise 3 (1 pt.)
Find a basis for the subspace of R3 .
V = [ x1 x2 x3 ]t ∈ R3 : x1 + x2 + x3 = 0
(you do not need to justify that V is a subspace).
[Hint: First find two linearly independent vectors in V (justify that they are linearly independent).
Then you have two options: either show directly that the two vectors span V , or argue that the
dimension of V must be 2, and so the two linearly independent vectors must form a basis. To show
that dim V = 2, use that there are 2 linearly independent vectors in it, and that it is not all of R3
(which you need to justify).]
Exercise 4 (2 pt.)
Consider the basis B = { ~v1 , ~v2 } of R2 , where ~v1 , ~v2 are the vectors:
1
1
~v1 =
, ~v2 =
1
4
(you do not need to justify that this is a basis).
1
(a) Find the matrix S such that ~x = S [~x]B for any ~x ∈ R2 . Also find its inverse S −1 , which is
the matrix such that [~x]B = S −1 ~x for any ~x ∈ R2 .
(b) Consider the linear transformation T : R2 → R2 given by the matrix
2 −1
A=
4 −3
Find [T (~v1 )]B and [T (~v2 )]B . Use your answer to find directly the matrix B of the linear
transformation T with respect to the basis B. Then find B using A and the matrices S and
S −1 from part (a). Compare your results.
(c) We denote by T n the composition
T
· · ◦ T} .
| ◦ ·{z
n times
Find the matrix associated to T 100 . (Your expression should not involve powers of matrices,
but may involve powers of numbers).
[Hint: For part (c), write A in terms of B, and note that powers of diagonal matrices are easy to
compute.]
Exercise 5 (1 pt.)
Let A, B, C ∈ Mn (R). Show the following:
(i) A is similar to itself.
(ii) If B is similar to A, then A is similar to B.
(iii) If C is similar to B, and B is similar to A, then C is similar to A.
(iv) If A, B are invertible and B is similar to A, then B −1 is similar to A−1 .
[Hint: For parts (iii) and (iv), you may use the identity (ST )−1 = T −1 S −1 for invertible matrices
S, T , which was proven in Exercise 5 of Homework 6.]
Exercise 6 (1 pt.)
Let B = {~v1 , . . . , ~vm } be a basis for a subspace V of Rn . Show the following:
(i) [~x + ~y ]B = [~x]B + [~y ]B for any ~x, ~y ∈ V , and
(ii) [c~x]B = c[~x]B for any ~x ∈ V , c ∈ R.
[Hint: Express ~x, ~y , ~x + ~y as linear combinations of the ~v1 , . . . , ~vm . How do these expressions relate
to [~x]B , [~y ]B , [~x + ~y ]B ?]
Exercise 7 (1 pt.)
Let T : Rn → Rm be a linear transformation. Show that:
(i) if n < m then the image of T cannot be equal to the whole space Rm , and
(ii) if n > m then the kernel of T cannot be trivial, i.e. ker T 6= {0}.
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[Hint: In both parts you need to use the Rank-Nullity Theorem for Linear Transformations. Argue
by contradiction: for (i), assume that the image of T is the whole space Rm . Then what is the
dimension of ker T ? Part (ii) is similar.]
Exercise 8 (Extra Credit, 1 pt.)
Let B and B 0 be two bases for Rn . Consider the function
T : Rn → Rn ,
T ([~x]B ) = [~x]B0 .
Show that T is a linear transformation and find the matrix associated to it.
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