Finding Basis for Subspaces of R4

In summary: In fact it is possible to prove that in this case, you can find a vector that is orthogonal to all the vectors in U+W. However, if W \cap U = {0}, then it is not possible to find such a vector since the vectors in U+W span the entire vector space R^4. In summary, we are given two subspaces W and U of R^4 and asked to find a homogenous system for W and a vector (x, y, z, t) that belongs to W. We can find the basis for W by using the given vectors and putting them into an extended matrix with (x, y, z, t) on the other side. We then perform elementary operations to
  • #1
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i am given 2 subspaces of R4
W=sp{(a-b,a+2b,a,b)|a,b[tex]\in[/tex]R}
U=sp{(1,0,1,1)(-6,8,-3,-2)}
and am asked to find:
a homogenic system for W- system for a vector (x,y,z,t) belonging to W

i see the basis for W is : a(1,1,1,0)+b(-1,2,0,1),, i put these vectors into an extended matrix with (x y z t) on the other side, and after a series of elementary operations, i get
x+t-z=0
y-z-2t=0

next i am asked to find
a basis for W+U and W[tex]\cap[/tex]U

forW[tex]\cap[/tex]U
i find a homogenic system for U, and compre it with the system i found for W which comes to
x+t-z=0
y-z-2t=0
y+8z-8t=0
3t-4z+x=0
and i come to
t=1.5z
z=z
y=4z
x=-0.5z
so for W[tex]\cap[/tex]U i get a basis (-0.5, 4, 1, 1.5)

for W+U i take the basis of each and check independace of all of them together, in which i get that all4 are independant, therefore the basis for W+U={(1,0,1,1)(-6,8,-3,-2)(1,1,1,0)(-1,2,0,1),}
if i perform elementary colum operations on them i can get to (1000)(0100)(0010)(0001), doesn't this mean that W+U is the whole vector space R4 ??

the final question is
find a vector other than the zero vector which is orthagonal to all the vectors in U+W
is this possible, since i found that W+U is the whole vector space R4 (supposing i was correct there)??
 
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  • #2
for W+U i take the basis of each and check independace of all of them together, in which i get that all4 are independant, therefore the basis for W+U={(1,0,1,1)(-6,8,-3,-2)(1,1,1,0)(-1,2,0,1),}
if i perform elementary colum operations on them i can get to (1000)(0100)(0010)(0001), doesn't this mean that W+U is the whole vector space R4 ??
Yes it does. In fact a certain theorem tells you that if you can find a set of n linearly independent vectors spanning R^n then that set is a basis for R^n. EDIT: I just checked it and I realized that the vectors are in fact not linearly independent. You need to re-check that part.

the final question is
find a vector other than the zero vector which is orthagonal to all the vectors in U+W
is this possible, since i found that W+U is the whole vector space R4 (supposing i was correct there)??
EDIT: As above the vectors are not linearly independent.
 
Last edited:
  • #3
If W [tex]\cap[/tex] U [tex]\neq[/tex] {0} then the set {basis of U} [tex]\cup[/tex] {basis of W} must be linearly dependent.
 

Related to Finding Basis for Subspaces of R4

1. What is a subspace of R4?

A subspace of R4 is a subset of the four-dimensional real vector space R4 that also satisfies the properties of a vector space. This means that a subspace must be closed under vector addition and scalar multiplication, and it must contain the zero vector.

2. How do you find the basis for a subspace of R4?

To find the basis for a subspace of R4, you must first determine the number of linearly independent vectors needed to span the subspace. Then, you can use those vectors as the basis for the subspace.

3. What is the dimension of a subspace of R4?

The dimension of a subspace of R4 is the number of linearly independent vectors needed to span the subspace. This is also known as the number of basis vectors for the subspace.

4. Can a subspace of R4 have more than one basis?

Yes, a subspace of R4 can have multiple bases. This is because there can be different sets of linearly independent vectors that can span the same subspace.

5. How can I check if a set of vectors is a basis for a subspace of R4?

To check if a set of vectors is a basis for a subspace of R4, you can use the properties of linear independence and spanning. Make sure that the vectors are linearly independent and that they span the entire subspace. If both conditions are met, then the set of vectors is a basis for the subspace.

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