Page 345 - 35Linear Algebra

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after one orbit the satellite will instead return to some other point Y ∈ R .
The engineer’s computations show that Y is related to X by a matrix

 1 
0 1
2
 1 
Y =  1 1  X .
 2 2 2
1 1 0
2
(a) Find all eigenvalues of the above matrix.
(b) Determine all possible eigenvectors associated with each eigenvalue.

Let us assume that the rule found by the engineers applies to all subsequent
orbits. Discuss case by case, what will happen to the satellite if the initial
mistake in its location is in a direction given by an eigenvector.

10. In this problem the scalars in the vector spaces are bits (0, 1 with 1+1 = 0).
k
The space B is the vector space of bit-valued, k-component column vectors.
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(a) Find a basis for B .
(b) Your answer to part (a) should be a list of vectors v 1 , v 2 , . . . v n . What
number did you find for n?
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(c) How many elements are there in the set B .
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(d) What is the dimension of the vector space B .
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(e) Suppose L : B → B = {0, 1} is a linear transformation. Explain why
specifying L(v 1 ), L(v 2 ), . . . , L(v n ) completely determines L.

(f) Use the notation of part (e) to list all linear transformations
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L : B → B .

How many different linear transformations did you find? Compare your
answer to part (c).

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(g) Suppose L 1 : B → B and L 2 : B → B are linear transformations,
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and α and β are bits. Define a new map (αL 1 + βL 2 ) : B → B by
(αL 1 + βL 2 )(v) = αL 1 (v) + βL 2 (v).

Is this map a linear transformation? Explain.
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(h) Do you think the set of all linear transformations from B to B is a
vector space using the addition rule above? If you answer yes, give a
basis for this vector space and state its dimension.

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