Subspaces of R4: W & U & Basis Calculation

In summary: So, in this case, finding the span of W and U will give you a basis for each subspace. And yes, writing the span is the same as writing one possible basis for the subspace.
  • #1
Dell
590
0
in this question i am given 2 subspaces of R4

W={(x1,x2,x3,x4)[tex]\in[/tex]R4|x1-x4=0}

U=sp{(1,2,0,-1),(2,3,1,-1),(1,-1,1,-1)}

and am asked to find
1] a basis and dimention for W
2] a basis for W+U
3] a basis for W[tex]\cap[/tex]U


1]
since i only have limitations on x1 and x4 i call x2=t x3=s x1=x4=q

therefore W=sp{(1,0,0,1),(0,1,0,0),(0,0,1,0)} and dimW=3

2]
to find a basis for W+U, i look for linearly independant vectors that span the space, so i set up a matrix to see which are combinations of the others,
i got W+U=sp{(1,0,0,0),(0,1,0,0),(0,0,1,0),(0,0,0,1)}

these 4 span the whole of R4 so does this mean that W+U = R4
3]
to find a basis of W[tex]\cap[/tex]U, i found a homogenic system for each subspace and compared the 2.
taking a random vector (a b c d) in the subspace, i get
for W==> -4a+b+3c-sd=0
for U==> a-d=0
from the combination i get -6a+b+3c=0
a=t b=6t-3s c=s d=t

and so i get W[tex]\cap[/tex]U=sp{(1,6,0,1)(0,-3,1,0)} dim=2
----------------------------------------

does this all look okay,
also, if i am asked to find a basis and i write the span, is it the same thing, or must i write just one possible basis in the span
 
Last edited:
Physics news on Phys.org
  • #2
?Yes, your answers look correct. To find a basis for a given subspace, you can find the span of the subspace and then any linearly independent set of vectors from this span will be a basis for the subspace.
 

What is a subspace of R4?

A subspace of R4 is a subset of R4 that satisfies the following properties: it contains the zero vector, it is closed under vector addition, and it is closed under scalar multiplication. In other words, it is a space within the larger space of R4 that behaves like a vector space.

What is the difference between W and U in subspaces of R4?

W and U refer to two different subspaces of R4. W is typically used to represent a specific subspace, while U is used as a general term for any subspace. It is common for W to be a subspace of U.

How do you determine if a set of vectors is a basis for a subspace of R4?

To determine if a set of vectors is a basis for a subspace of R4, you must check if the vectors are linearly independent and if they span the subspace. This means that none of the vectors can be written as a linear combination of the others, and that they can be used to create any vector within the subspace. If both of these conditions are met, then the set of vectors is a basis for the subspace.

Can a subspace of R4 have more than one basis?

Yes, a subspace of R4 can have more than one basis. This is because there can be multiple sets of linearly independent vectors that span the same subspace. However, all bases for a given subspace will have the same number of vectors, known as the dimension of the subspace.

How do you calculate the basis of a subspace of R4?

To calculate the basis of a subspace of R4, you must first find a set of linearly independent vectors that span the subspace. This can be done by setting up a system of equations and solving for the variables. Once you have a set of linearly independent vectors, you can check if they span the subspace. If they do, then they are a basis for the subspace. If not, you may need to add or remove vectors until you have a basis.

Similar threads

  • Calculus and Beyond Homework Help
Replies
4
Views
1K
  • Calculus and Beyond Homework Help
Replies
0
Views
408
  • Calculus and Beyond Homework Help
Replies
8
Views
1K
  • Calculus and Beyond Homework Help
Replies
14
Views
514
  • Calculus and Beyond Homework Help
Replies
9
Views
2K
  • Calculus and Beyond Homework Help
Replies
7
Views
1K
  • Calculus and Beyond Homework Help
Replies
2
Views
244
  • Calculus and Beyond Homework Help
Replies
8
Views
535
  • Calculus and Beyond Homework Help
Replies
14
Views
1K
  • Calculus and Beyond Homework Help
Replies
10
Views
2K
Back
Top