Basis is finite set of vectors that are linearly independant

When finding the basis for the row space, the answer is a set of vectors that span the row space, but it is not the span of those vectors. This is because the basis should be a linearly independent set of vectors, while the span may contain linearly dependent vectors. In summary, the basis of a vector space is a finite set of linearly independent vectors that span the space, but when finding the basis for a specific vector space, the answer may not be the span of those vectors as the basis should be a linearly independent set.
  • #1
squenshl
479
4
I know that the basis is finite set of vectors that are linearly independant and SPANS that set. But why is when you find the basis for the row space for example the answer is {[u1,u2,...,um}} and not span{[u1,u2,...,um}]. I got this wrong in a test. I don't see why eventhough the definition states that it spans that set.
 
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  • #2


If {u1, u2, ..., um} spans the vector space V, then span{u1, u2, ..., um} = V, right? And V is obviously not a basis for V.
 
  • #3


A basis is a set of vectors. The span of a set of vectors is a vector space, not a basis.
 

1. What is a basis?

A basis is a set of vectors that can be used to uniquely describe any other vector in a vector space through linear combinations.

2. What does it mean for a set of vectors to be linearly independent?

A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the other vectors. In other words, no vector in the set is redundant and each vector adds new information to the set.

3. Why is it important for a basis to be finite?

A finite basis ensures that the vector space has a finite dimension, which makes it easier to work with mathematically. It also allows for simpler and more efficient calculations in applications such as solving systems of linear equations.

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

Yes, a vector space can have multiple bases. However, all bases for a given vector space must have the same number of vectors, known as the dimension of the vector space.

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

To determine if a set of vectors is a basis for a given vector space, you can check if the set is linearly independent and if it spans the entire vector space. If both of these conditions are met, then the set is a basis for the vector space.

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