Showing a set is a basis for a vector space

• I
• penroseandpaper
In summary, to find a basis for a set of vectors in a linear space, first determine if the set is linearly independent, then use linear combinations of the vectors in the set to create a basis for the space.
penroseandpaper
If I'm given a set of four vectors, such as A={(0,1,4,2),(1,0,0,1)...} and am given another set B, whose vectors are given as a form such as (x, y, z, x+y-z) all in ℝ, what steps are needed to show A is a basis of B?

I have calculated another basis of B, and found I can use linear combinations of the vectors in this basis to make each of the four vectors in A. But I'm not sure if I can use that as proof or if it means anything.

No answers being sought, simply a checklist of steps to take. The set notation including (x, y, z, x+y-z) has thrown me.

Penn

##A## is a basis of ##B## makes only sense, if ##B## is a linear space. So I assume we have to decide whether ##A## is a basis for the linear span ##\operatorname{lin}B## of the vectors in ##B##. Let's call this vector space ##\mathcal{B}=\operatorname{lin}B## in order to distinguish it from the set of vectors ##B## which you used.

The necessary steps are as follows:
1. ##A \subseteq \mathcal{B}##, i.e. each vector ##\vec{a}\in A## must have a linear combination of vectors from ##B##, i.e. ##\vec{a}=\sum_{i=1}^k \lambda_i\vec{b}_i## with ##\lambda_i\in \mathbb{R},\vec{b}_i\in B##.
2. ##A## must span ##B##, i.e. the other way around must also hold: each vector ##\vec{b}\in B## must have a linear combination of vectors from ##A##, i.e. ##\vec{b}=\sum_{i=1}^k \mu_i\vec{a}_i## with ##\mu_i\in \mathbb{R},\vec{a}_i\in A##.
3. ##A=\{\vec{a}_1,\ldots,\vec{a}_n\}## must be linearly independent, i.e. from ##\vec{0}=\sum_{i=1}^n x_i\vec{a}_i## must follow, that the equation can only hold if ##x_1=\ldots=x_n=0.##

Are you sure about the problem? B is 3 dimensional, 4 vectors cannot be a basis.

1. What does it mean for a set to be a basis for a vector space?

A basis for a vector space is a set of vectors that can be used to represent any vector in that space through linear combinations. This means that any vector in the space can be written as a unique combination of the basis vectors, and the basis vectors are linearly independent.

2. How do you show that a set is a basis for a vector space?

To show that a set is a basis for a vector space, you need to prove two things: linear independence and spanning. This means that the set of vectors must be linearly independent, meaning no vector can be written as a linear combination of the other vectors in the set. Additionally, the set must span the entire vector space, meaning that every vector in the space can be written as a linear combination of the basis vectors.

3. Can a set with more than the dimension of the vector space be a basis?

No, a basis for a vector space must have the same number of vectors as the dimension of the vector space. If a set has more vectors than the dimension of the space, it cannot be a basis because it would contain redundant vectors, and therefore would not be linearly independent.

4. Is a basis for a vector space unique?

Yes, a basis for a vector space is unique. This means that any two bases for the same vector space will have the same number of vectors, and any vector in the space can be written as a unique combination of the basis vectors. However, there can be different sets of vectors that are also bases for the same vector space.

5. Can a set with linearly dependent vectors be a basis for a vector space?

No, a basis for a vector space must consist of linearly independent vectors. If a set contains linearly dependent vectors, it means that at least one vector in the set can be written as a linear combination of the other vectors, making it redundant. Therefore, it cannot be a basis for the vector space.

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