Showing a set is a basis for a vector space

  • #1
penroseandpaper
21
0
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
 

Answers and Replies

  • #2
fresh_42
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##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.##
 
  • #3
martinbn
Science Advisor
3,048
1,386
Are you sure about the problem? B is 3 dimensional, 4 vectors cannot be a basis.
 

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