Finding a basis given part of that basis

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In summary, for both (a) and (b), the key is to find a vector that is orthogonal to the span of the given vectors. For (a), you can solve the system S'*x=0, and for (b), you can "straighten" the 2x2 matrices into 4x1 vectors and find the orthogonal vector in the same way.
  • #1
skyturnred
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Homework Statement



This is the question on my assignment:

In each case below, given a vector space V , find a basis B for V containing the linearly independent set S ⊂ B.

It has a bunch of different cases but I think that if you help me with the following two, I will learn enough to do the others. The first case is the following:

(a) V = R[itex]^{4}[/itex], S = {(1,0,0,1),(0,1,1,0),(2,1,1,1)}.

and the next case is:

(b) V = M2×2 = the vector space of all 2 × 2 matrices, and

S= [1, 1; 1, 0] [0, 1; 1, 1] [1, 0; 1, 1]

Homework Equations





The Attempt at a Solution



My problem with BOTH cases is this: I only know how to find a basis given a bunch of vectors that form a span (in other words, I know how to find the linearly dependent ones and kick them out of the equation). But I DONT understand how to find the missing parts of the basis given what the basis is SUPPOSED to span. Can someone please walk me through this?

And my SECOND problem is with case (b): I cannot visualize how matrices can span something. I understand vectors, but not matrices. And since I can't understand it, I can't approach it to find the basis.

Thanks SO much in advance for your help!
 
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  • #2
For (a), what you can do is solve the system S'*x=0 for a nonzero solution. In other words, you want to find a vector that are orthogonal to the span of columns of S, i.e., in the nullspace of S'. You may start with a random vector, project onto span{S} and subtract the projection, if you're lucky the remainder is nonzero, either the remainder or the original random vector is linearly independent.

For (b), all you have to do is "straighten" the 2x2 matrix into a 4x1 vector. As linear spaces they are isomorphic.
 
  • #3
OK, I am a bit confused. So solving S'*x-0, S' is the matrix whose columns are the vectors of S right? So when I do that I will be finding the nullspace of that matrix..

When I do this, I get x1, x2, and x3 all equal to 0. Is this right? I'm sorry for my thick headedness.. but linear algebra is by far my weakest class.
 
  • #4
skyturnred said:
OK, I am a bit confused. So solving S'*x-0, S' is the matrix whose columns are the vectors of S right? So when I do that I will be finding the nullspace of that matrix..

When I do this, I get x1, x2, and x3 all equal to 0. Is this right? I'm sorry for my thick headedness.. but linear algebra is by far my weakest class.

No worries, I should've been more explicit. S' is the transpose of S, where S is the 4x3 matrix whose columns are those vectors, so S' is 3x4. Your solution should be a 4x1 vector (x1,x2,x3,x4)≠0, it is orthogonal and therefore independent to rows of S', i.e., columns of S, i.e., your original vectors.
 

Related to Finding a basis given part of that basis

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 expressed as a linear combination of the basis vectors.

2. Why is it important to find a basis for a vector space?

Finding a basis for a vector space is important because it helps us understand the structure of the vector space and allows us to perform operations such as finding linear combinations and determining linear independence. It also simplifies calculations and makes it easier to solve problems in linear algebra.

3. How do you find a basis given part of that basis?

To find a basis given part of that basis, you can use the concept of linear independence. Start with the given basis vectors and add additional vectors one by one. Each time you add a new vector, check if it is linearly independent from the existing set of vectors. If it is, then it can be added to the basis. Continue this process until you have a set of linearly independent vectors that span the vector space.

4. Can there be more than one basis for a given vector space?

Yes, there can be more than one basis for a given vector space. In fact, any set of linearly independent vectors that span the vector space can be considered a basis. However, all bases for a given vector space will have the same number of vectors, known as the dimension of the vector space.

5. How can finding a basis be useful in real-world applications?

Finding a basis can be useful in real-world applications such as data analysis and machine learning. In these fields, data is often represented as vectors in high-dimensional vector spaces. By finding a basis, we can reduce the dimensionality of the data and make it easier to analyze and interpret. Bases are also used in computer graphics, where they are used to represent and manipulate 3D objects.

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