Vector Subspaces

[Jaglowsd]

Let v be a vector in Rⁿ . Define W to be the set of vectors w in Rⁿ such that w (dot) v = 0. Show that W is a vector space of Rⁿ.

I know that the following three properties need to hold in order for W to be in the subspace:

1) Vector addition of vectors u,v
2) Scalar multiplication of a real number a, and u
3) A vector space has to have a zero vector

1) Since v,w are both in Rⁿ I thought I could try,

w (dot) v = 0
since, w is a vector such that w (dot) v = 0,
w + v = 0 where w is [w (dot) v] then,
[w (dot) v] + v = 0
0 + v = 0, which would be false unless v=0.
I don't think I am going about this correctly.

A little bit of guidance would be very useful! Thanks

I realize that the notation is a little confusing with two lower case w's, but that is how it is stated in the text.

 

[pka]

Let v be a vector in Rⁿ . Define W to be the set of vectors w in Rⁿ such that w (dot) v = 0. Show that W is a vector space of Rⁿ.
I know that the following three properties need to hold in order for W to be in the subspace:
1) Vector addition of vectors u,v
2) Scalar multiplication of a real number a, and u
3) A vector space has to have a zero vector

To be clear, you are showing that \(\displaystyle \bf W\) is a subspace of \(\displaystyle \bf R^n\).

Note that \(\displaystyle \bf O\cdot v =0\). And \(\displaystyle \bf (au+w)\cdot v=a(u\cdot v)+w\cdot v\)

 

[HallsofIvy]

Let v be a vector in Rⁿ . Define W to be the set of vectors w in Rⁿ such that w (dot) v = 0. Show that W is a vector space of Rⁿ.

I know that the following three properties need to hold in order for W to be in the subspace:

1) Vector addition of vectors u,v
2) Scalar multiplication of a real number a, and u
3) A vector space has to have a zero vector

1) Since v,w are both in Rⁿ I thought I could try,

w (dot) v = 0
since, w is a vector such that w (dot) v = 0,
w + v = 0 where w is [w (dot) v] then,

This makes no sense. You said w was a vector. But then you cannot say "where w is w (dot) v" because w (dot) v is not a vector.
And, given that w (dot) v= 0, it does NOT follow that "w+ v= 0".

[w (dot) v] + v = 0

And this makes no sense. You cannot add a number (w (dot) v) and a vector (v).

0 + v = 0, which would be false unless v=0.
I don't think I am going about this correctly.

You aren't! I think you have completely misunderstood the question.

A little bit of guidance would be very useful! Thanks

I realize that the notation is a little confusing with two lower case w's, but that is how it is stated in the text.

As you say, in order to be a vector space a subset of vectors must be closed under vector addition. Suppose x and y are two vectors in the set. Then we must have x (dot) v= 0 and y (dot) v= 0. What can you say about (x+ y) (dot) v?

As you say, in order to be a vector space, a subset of vectors must be closed under scalar multiplication. Suppose x is a vector in the set and a is a scalar. Then we must have x (dot) v= 0. What can you say about (ax) (dot) v?

As you say, in order to be a vector space, a subset of vectors must contain the 0 vector. What can you say about 0 (dot) v?