Find a basis for a set S of R4

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In summary, a basis for a set of vectors in R4 is a set of linearly independent vectors that span the entire vector space of R4. To find a basis, you can use the Gauss-Jordan elimination method to reduce the set of vectors into a reduced row echelon form. This is important because it simplifies calculations and allows for a better understanding of the vector space. A set of vectors in R4 can have multiple bases, but all bases will have the same number of vectors. To determine if a set of vectors is a basis, you can check if the vectors are linearly independent and span the entire vector space of R4 using methods such as Gauss-Jordan elimination or calculating the determinant.
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ElijahRockers
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Homework Statement



Find a basis for the subspace S of vectors (A+B, A-B+2C, B, C) in R4

What is the dimension of S?

The Attempt at a Solution



Do I just plug in varying values for A B and C to create four vectors, and see if they are linearly independent? If they are then I've found a basis?
 
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  • #2
Making any two of A, B & C zero results in a vector in S. How many such vectors are there?
 
  • #3
Or, similar to voko's observation write it like this:$$
A(1,1,0,0)+B(1,-1,1,0)+C(0,2,0,1)$$and see if that helps.
 

FAQ: Find a basis for a set S of R4

What is a basis for a set of vectors in R4?

A basis for a set of vectors in R4 is a set of linearly independent vectors that span the entire vector space of R4. In other words, any vector in R4 can be written as a unique linear combination of the basis vectors.

How do you find a basis for a set of vectors in R4?

To find a basis for a set of vectors in R4, you can use the Gauss-Jordan elimination method to reduce the set of vectors into a reduced row echelon form. The nonzero rows in this form will form the basis for the set of vectors.

Why is it important to find a basis for a set of vectors in R4?

Finding a basis for a set of vectors in R4 is important because it allows us to represent any vector in R4 using a linear combination of a few basis vectors. This simplifies calculations and makes it easier to understand the vector space.

Can a set of vectors in R4 have more than one basis?

Yes, a set of vectors in R4 can have multiple bases. This is because there are infinitely many ways to choose a set of linearly independent vectors that span a vector space. However, all bases for a given set of vectors will have the same number of vectors.

How can you determine if a set of vectors in R4 is a basis?

A set of vectors in R4 is a basis if it satisfies two conditions: 1) the vectors are linearly independent, meaning that none of the vectors can be written as a linear combination of the others, and 2) the vectors span the entire vector space of R4. You can check these conditions by using the Gauss-Jordan elimination method or by calculating the determinant of the matrix formed by the vectors.

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