Page 333 - 35Linear Algebra
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6. Suppose λ is an eigenvalue of the matrix M with associated eigenvector v.
Is v an eigenvector of M k (where k is any positive integer)? If so, what
would the associated eigenvalue be?
Now suppose that the matrix N is nilpotent, i.e.
k
N = 0
for some integer k ≥ 2. Show that 0 is the only eigenvalue of N.
!
3 −5
12
7. Let M = . Compute M . (Hint: 2 12 = 4096.)
1 −3
8. The Cayley Hamilton Theorem: Calculate the characteristic polynomial
a b
P M (λ) of the matrix M = . Now compute the matrix polynomial
c d
P M (M). What do you observe? Now suppose the n×n matrix A is “similar”
to a diagonal matrix D, in other words
A = P −1 DP
for some invertible matrix P and D is a matrix with values λ 1 , λ 2 , . . . λ n
along its diagonal. Show that the two matrix polynomials P A (A) and P A (D)
are similar (i.e. P A (A) = P −1 P A (D)P). Finally, compute P A (D), what can
you say about P A (A)?
9. Define what it means for a set U to be a subspace of a vector space V .
Now let U and W be non-trivial subspaces of V . Are the following also
subspaces? (Remember that ∪ means “union” and ∩ means “intersection”.)
(a) U ∪ W
(b) U ∩ W
3
In each case draw examples in R that justify your answers. If you answered
“yes” to either part also give a general explanation why this is the case.
10. Define what it means for a set of vectors {v 1 , v 2 , . . . , v n } to (i) be linearly
independent, (ii) span a vector space V and (iii) be a basis for a vector
space V .
3
Consider the following vectors in R
−1 4 10
5
u = −4 , v = , w = 7 .
3 0 h + 3
3
For which values of h is {u, v, w} a basis for R ?
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