Finding basis for subspace

In summary, to determine a basis for the subspace of R^n spanned by the given set of vectors, {(1,-1,2),(5,-4,1),(7,-5,-4)}, we need to find the linear combinations of the vectors that equal the zero vector. After performing the necessary calculations, we can see that any two of the given vectors form a basis for their span, which is a two-dimensional subspace of R3. The use of rowspace and colspace is not applicable in this situation.
  • #1
snoggerT
186
0
use rowspace/colspace to determine a basis for the subspace of R^n spanned by the given set of vectors:

{(1,-1,2),(5,-4,1),(7,-5,-4)}


*note: the actual instructions are to use the ideas in the section to determine the basis, but the only two things learned in the section are rowspace and colspace.

The Attempt at a Solution



- I thought you could just find the rowspace, and that would be a subspace of R^n, but the answer in the back of the book isn't at all the same. How would you go about solving this problem?
 
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  • #2
snoggerT said:
use rowspace/colspace to determine a basis for the subspace of R^n spanned by the given set of vectors:

{(1,-1,2),(5,-4,1),(7,-5,-4)}

- I thought you could just find the rowspace, and that would be a subspace of R^n, but the answer in the back of the book isn't at all the same. How would you go about solving this problem?

What's the definition of basis in your book?
 
  • #3
Since there is no matrix here, I cannot see how it can have anything to do with "rowspace" of "columnspace"! The "rowspace" is, after all, the space spanned by the rows of a matrix (thought of as vectors) and, of course, the "columnspace" is the spanned by the columms- but you have no matrix here.

Can we assume that, since this problem asks about a basis, you know what a "basis" is? Was that in a previous section so you think you shouldn't use the definition? Also, since each of the vectors given is in R3, we are assuming that n= 3.

A set of vectors always spans some vector space. It that set is also [bindependent[/b] then it is a basis for the space. If a set is not independent then one or more of the vectors can be written as a linear combination of the others. Get rid of those and you have a basis.

"Independent" means that a linear combination equal to the 0 vector must have all coefficients 0. Here, that means we must look at a(1,-1,2)+ b(5,-4,1)+ c(7,-5,-4)= (0, 0, 0) so we must have a+ 5b+ 7c= 0, -a- 4b- 5c= 0, and 2a+ b- 4c= 0. If we multiply the second equation by 2 and add to the third, we get -7b- 14c= 0 or b= -2c. If add the first and second equations, we get b+ 2c= 0: again b= -2c. Putting b= -2c into the second equation, -a- 4b+ 10b= -a+ 6b= 0 so a= 6b. Taking b= 1, a= 6, c= -2 and we have 6(1, -1, 2)+ (5, -4, 1)- 2(7, -5, -4)= (0,0,0). We can take anyone of those to the right side and divide by its coefficient to see that it can be replace by a linear combination of the other 2. Any two of the given vectors forms a basis for their span (and show that the span is a two dimension subspace of R3).

NateTG got in just ahead of me! I need to learn to type faster!
 
  • #4
HallsofIvy said:
Since there is no matrix here, I cannot see how it can have anything to do with "rowspace" of "columnspace"! The "rowspace" is, after all, the space spanned by the rows of a matrix (thought of as vectors) and, of course, the "columnspace" is the spanned by the columms- but you have no matrix here.

Can we assume that, since this problem asks about a basis, you know what a "basis" is? Was that in a previous section so you think you shouldn't use the definition? Also, since each of the vectors given is in R3, we are assuming that n= 3.

A set of vectors always spans some vector space. It that set is also [bindependent[/b] then it is a basis for the space. If a set is not independent then one or more of the vectors can be written as a linear combination of the others. Get rid of those and you have a basis.

"Independent" means that a linear combination equal to the 0 vector must have all coefficients 0. Here, that means we must look at a(1,-1,2)+ b(5,-4,1)+ c(7,-5,-4)= (0, 0, 0) so we must have a+ 5b+ 7c= 0, -a- 4b- 5c= 0, and 2a+ b- 4c= 0. If we multiply the second equation by 2 and add to the third, we get -7b- 14c= 0 or b= -2c. If add the first and second equations, we get b+ 2c= 0: again b= -2c. Putting b= -2c into the second equation, -a- 4b+ 10b= -a+ 6b= 0 so a= 6b. Taking b= 1, a= 6, c= -2 and we have 6(1, -1, 2)+ (5, -4, 1)- 2(7, -5, -4)= (0,0,0). We can take anyone of those to the right side and divide by its coefficient to see that it can be replace by a linear combination of the other 2. Any two of the given vectors forms a basis for their span (and show that the span is a two dimension subspace of R3).

NateTG got in just ahead of me! I need to learn to type faster!

- Can you explain to me how plugging b=-2c back into the 2nd equation gives you -a-4b+10b?
 

What is a subspace and why is it important in linear algebra?

A subspace is a subset of a vector space that satisfies the same axioms and operations as the original vector space. It is important in linear algebra because it allows us to simplify complex vector spaces and focus on specific properties and behaviors. Subspaces also help us understand the underlying structure of a vector space.

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

To determine if a set of vectors form a basis for a subspace, we need to check if the set is linearly independent and spans the subspace. This means that none of the vectors can be written as a linear combination of the others, and that every vector in the subspace can be expressed as a linear combination of the set of vectors.

Can a subspace have more than one basis?

Yes, a subspace can have more than one basis. As long as the set of vectors is linearly independent and spans the subspace, it can be considered a basis for that subspace. This means that there can be infinitely many bases for a given subspace.

What is the difference between a basis and a spanning set?

A basis is a set of vectors that are linearly independent and span the entire vector space, while a spanning set is a set of vectors that can generate the entire vector space but may not be linearly independent. A basis is a minimal spanning set, meaning it is the smallest set of vectors that can span the vector space.

How do you find a basis for a subspace?

To find a basis for a subspace, we need to find a set of linearly independent vectors that span the subspace. This can be done by solving a system of linear equations or using other methods such as Gaussian elimination or Gram-Schmidt process. We can also use the concept of dimension to determine the number of basis vectors needed for a given subspace.

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