Finding Basis of W: Linear Independence of X1,X2,X3,X4

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In summary, the basis of the vector space W is formed by taking any three linearly independent vectors from the system of equations x1 = -2x2 - x4, x2 = 1x2, x3 = 1x3, x4 = 1x4. The dimension of W is 3.
  • #1
Dell
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given these 4 vectors X1..4 in vector space W

W=sp{(X1,X2,X3X4)[tex]\in[/tex]r4| X1+2X3+X4=0}

Find the basis of W,

what i did was--
for these to be the basis of W they need to span W and be independant of one another,-
since X1+2X3+X4=0 it is clear that these 3 are not independant, since X1+2X3=-X4 etc..

does this mean that dimW=3 and i can take any combination of 3 of the 4 x's for my basis as long as X2 is one of them?
 
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  • #2
x1, x2, x3, and x4 aren't vectors; they are coordinates of an arbitrary vector in R4. The problem could just have easily been stated in terms of x, y, z, and w.

From the definition of the set W, I can tell by inspection that (1, 1, 0, -3) is in the set, and I got this by substituting values into the equation x1 + 2x2 + x4 = 0.

You can characterize all of the vectors in W just from the equation and three other obvious equations:
x1 = - 2x2 - x4
x2 = 1x2
x3 = 1x3
x4 = 1x4

If you look at this system of equations for a while, you might see some vectors lurking in there. You might even be able to convince yourself that they are linearly independent and span W.
 
  • #3
thanks, so the dimention would be 1
and the basis would be that vector (made up of)
x1 = - 2x2 - x4
x2 = 1x2
x3 = 1x3
x4 = 1x4
 
  • #4
Dell said:
thanks, so the dimention would be 1
and the basis would be that vector (made up of)
x1 = - 2x2 - x4
x2 = 1x2
x3 = 1x3
x4 = 1x4

No, the dimension is not 1. Notice that for each vector (x1, x2, x3, x4) there are three parameters that can be set. Does that tell you something?
 
  • #5
right, x2 x3 and x4 are all independant,
dim=3
 
  • #6
Dell said:
right, x2 x3 and x4 are all independant,
dim=3
No on x2, x3, and x4. Yes on the dimension.

As I said earlier, x2, x3, and x4 are not vectors, so it doesn't make any sense to describe them as linearly dependent or linearly independent.

You should be able to get three lin. independent vectors out of the system of equations I wrote in a previous post.
 

Related to Finding Basis of W: Linear Independence of X1,X2,X3,X4

1. What is the definition of "basis" in linear algebra?

A basis is a set of vectors that can be used to represent any other vector in a vector space through linear combinations. In other words, a basis is a set of vectors that are linearly independent and span the entire space.

2. How do you determine if a set of vectors is linearly independent?

A set of vectors is linearly independent if none of the vectors in the set can be written as a linear combination of the other vectors. This means that the coefficients of the linear combination must all be equal to zero.

3. Can a set of vectors that are linearly dependent still form a basis?

No, a basis must consist of linearly independent vectors. If a set of vectors is linearly dependent, it means that at least one vector can be written as a linear combination of the others, which violates the definition of a basis.

4. How do you find the basis of a given vector space?

To find the basis of a vector space, you can start by identifying a set of linearly independent vectors that span the space. Then, you can check if any additional vectors can be added to the set to create a basis. This process may involve using techniques such as Gaussian elimination or finding the null space of a matrix.

5. What is the relationship between the basis and the dimension of a vector space?

The dimension of a vector space is equal to the number of vectors in its basis. This means that the basis is a fundamental concept in understanding the dimension of a vector space. Additionally, any set of linearly independent vectors in a vector space that has the same number of vectors as the basis is also considered a basis for that space.

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