Page 406 - 35Linear Algebra

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Worked Proof

Here we will work through a quick version of the proof of Theorem 10.1.1. Let
P i
{v i } denote a set of linearly dependent vectors, so i c v i = 0 where there
k
exists some c 6= 0. Now without loss of generality we order our vectors such
1
that c 6= 0, and we can do so since addition is commutative (i.e. a+b = b+a).
Therefore we have
n
X i
1
c v 1 = − c v i
i=2
n i
X c
v 1 = − v i
c 1
i=2
i
and we note that this argument is completely reversible since every c 6= 0 is
i
invertible and 0/c = 0.

Hint for Review Problem 1

Lets first remember how Z 2 works. The only two elements are 1 and 0. Which
means when you add 1 + 1 you get 0. It also means when you have a vector ~v ∈ B n
and you want to multiply it by a scalar, your only choices are 1 and 0. This
is kind of neat because it means that the possibilities are finite, so we can
look at an entire vector space.
Now lets think about B 3 there is choice you have to make for each co-
ordinate, you can either put a 1 or a 0, there are three places where you
3
have to make a decision between two things. This means that you have 2 = 8
3
possibilities for vectors in B .
When you want to think about finding a set S that will span B 3 and is
linearly independent, you want to think about how many vectors you need. You
3
will need you have enough so that you can make every vector in B using linear
combinations of elements in S but you don’t want too many so that some of
them are linear combinations of each other. I suggest trying something really
simple perhaps something that looks like the columns of the identity matrix
For part (c) you have to show that you can write every one of the elements
as a linear combination of the elements in S, this will check to make sure S
3
actually spans B .
For part (d) if you have two vectors that you think will span the space,
you can prove that they do by repeating what you did in part (c), check that
every vector can be written using only copies of of these two vectors. If you
don’t think it will work you should show why, perhaps using an argument that
counts the number of possible vectors in the span of two vectors.

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