Linear Algebra Finding Basis for space.

In summary, the conversation discussed finding a basis for a subspace of R^4, W = {(x,y,z,w) : x+z=0, 2y+w=0}, and determining its dimension. The attempted solution involved finding two linearly independent vectors, (-1,0,1,0) and (0,-1/2,0,1), and questioning if using the variables x and y instead of z and w would have been correct. The experts confirmed that the obtained vectors were a basis for W and its dimension is 2. The person was advised to talk to their teacher about the marks received for this problem.
  • #1
jordan123
16
0

Homework Statement


Says, The set W = {(x,y,z,w) : x+z=0, 2y+w=0} is a subspace of R^4. Find a basis for W, and state the dimension.

The Attempt at a Solution



What I did:

W= {(-z,-w/2,z,w): z,w are in R}
= {z(-1,0,1,0) + w(0,-1/2, 0, 1)}
= span {(-1,0,1,0), (0,-1/2,0,1)}

(-1,0,1,0), (0,-1/2,0,1) is LI, because not a scalar multiple. Basis has dimension 2.

I got this wrong.

What is wrong? I guess I could have used the other variables, x and y, instead of z,w Would that have been correct? Or should I go to my teacher and get some marks, brought me down 7 percent?

Thanks
 
Last edited:
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  • #2
This seems right.
I used w= -2y and got [ 0 1 0 -2] which is essentially the same basis. i guess u used y=-w/2.
And ur first base is right
 
  • #3
fireb said:
This seems right.
I used w= -2y and got [ 0 1 0 -2] which is essentially the same basis. i guess u used y=-w/2.
And ur first base is right


Yeah, I just used x=-z and y=-(w/2)
 
  • #4
The vectors you got are a basis for W, and the dimension of this subspace of R^4 is 2. You should definitely talk to your teacher and ask why you lost points for this problem. I got exactly the same vectors you did.
 
  • #5
Mark44 said:
The vectors you got are a basis for W, and the dimension of this subspace of R^4 is 2. You should definitely talk to your teacher and ask why you lost points for this problem. I got exactly the same vectors you did.

Awesome, thanks! I will. And the odd thing is, on the midterm it is marked right and then crossed out and marked wrong.
 

1. What is a basis in linear algebra?

A basis in linear algebra is a set of linearly independent vectors that span a vector space. This means that any vector in the vector space can be written as a linear combination of the basis vectors.

2. How do you find a basis for a vector space?

To find a basis for a vector space, you need to find a set of linearly independent vectors that span the space. This can be done by solving a system of linear equations or by using the Gaussian elimination method. Once you have a set of linearly independent vectors, you can check if they span the space by making sure that every vector in the space can be written as a linear combination of the basis vectors.

3. What is the importance of finding a basis for a vector space?

Finding a basis for a vector space is important because it allows us to understand the structure of the space and perform calculations and transformations on vectors in the space. It also helps us to solve systems of linear equations and find solutions to linear systems.

4. Can a vector space have more than one basis?

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

5. How do you know if a set of vectors is a basis for a vector space?

To know if a set of vectors is a basis for a vector space, you can check if the vectors are linearly independent and if they span the space. This can be done by solving a system of linear equations or by using the Gaussian elimination method. If the vectors are linearly independent and span the space, then they are a basis for the vector space.

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