Page 404 - 35Linear Algebra

404 Movie Scripts

Here we have taken the subspace W to be a plane through the origin and U to
be a line through the origin. The hint now is to think about what happens when
you add a vector u ∈ U to a vector w ∈ W. Does this live in the union U ∪ W?
For the second part, we take a more theoretical approach. Lets suppose
0
that v ∈ U ∩ W and v ∈ U ∩ W. This implies
0
v ∈ U and v ∈ U .
So, since U is a subspace and all subspaces are vector spaces, we know that
the linear combination
0
αv + βv ∈ U .
Now repeat the same logic for W and you will be nearly done.

G.9 Linear Independence

Worked Example

This video gives some more details behind the example for the following four
3
3
vectors in R Consider the following vectors in R :
       
4 −3 5 −1
v 1 =  −1  , v 2 =  7  , v 3 =  12  , v 4 =  1  .
3 4 17 0
The example asks whether they are linearly independent, and the answer is
3
immediate: NO, four vectors can never be linearly independent in R . This
vector space is simply not big enough for that, but you need to understand the

404