Finding the dimension of the span ##u_1-u_2, ...., u_n-u_1##?

In summary, to determine the dimension of the span of ##\textbf{u}_1-\textbf{u}_2, \textbf{u}_2-\textbf{u}_3,...,\textbf{u}_n-\textbf{u}_1##, one must find the number of vectors that make up the basis of the span. This number is equal to the dimension of the span. To find the basis, one must check the linear independence of the vectors by setting the coefficients of the linear combinations to zero. If all coefficients are zero, the vectors are linearly dependent. To proceed, one should eliminate one vector from the dependent set and check for independence again. This process can be repeated until the dimension
  • #1
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If ##\textbf{u}_1,...,\textbf{u}_n## form a basis in a linear space, how does one determine the dimension of the span ##\textbf{u}_1-\textbf{u}_2, \textbf{u}_2-\textbf{u}_3,...,\textbf{u}_n-\textbf{u}_1##? Since ##\textbf{u}_1,...,\textbf{u}_n## form a basis, they're linearly independent. If one finds the number of vectors that make up the basis of the span ##\textbf{u}_1-\textbf{u}_2,...,\textbf{u}_n-\textbf{u}_1##, then that number is also its dimension. To find the basis of the span, I check the linear independence of the vectors:

##x_1(\textbf{u}_1-\textbf{u}_2)+x_2(\textbf{u}_2-\textbf{u}_3) +...+x_n(\textbf{u}_n-\textbf{u}_1)=0\qquad(1)##​
Rearranging terms:

##(x_1-x_n)\textbf{u}_1+(x_2-x_1)\textbf{u}_2+...+(x_n-x_{n-1})\textbf{u}_n=0##​

From the linear independence of the vectors ##\textbf{u}_1,...,\textbf{u}_n##, ##(x_1-x_n)=(x_2-x_1)=...=(x_n-x_{n-1})=0##, which means that ##x_1=x_2=...=x_n=k## for some ##k\in\mathbf{R}##. (1) can thus be written ##k((\textbf{u}_1-\textbf{u}_2)+(\textbf{u}_2-\textbf{u}_3) +...+(\textbf{u}_n-\textbf{u}_1))=k(\textbf{0})=0##. Thus they're all linear dependent. How does one proceed from here?
 
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  • #2
schniefen said:
How does one proceed from here?
Eliminate one vector from the dependent set and check again.

Loosely speaking, eliminating ##(u_k-u_{k+1})## from the set and checking for independence is like setting ##x_k=0## in all of your steps which then makes all coefficients zero, but you should repeat the process and see.
 
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  • #3
isn't the sum of your vectors zero? so their span has dim ≤ n-1. indeed it seems to equal n-1 since if you throw in un, you seem to get everything. but i did not check it on paper.
 
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Related to Finding the dimension of the span ##u_1-u_2, ...., u_n-u_1##?

1. What does it mean to find the dimension of the span of vectors?

Finding the dimension of the span of vectors means determining the number of linearly independent vectors that can be created by taking linear combinations of the given vectors. In other words, it is finding the maximum number of vectors that can be generated using the given vectors.

2. Why is it important to find the dimension of the span of vectors?

Knowing the dimension of the span of vectors is important in various areas of mathematics and science, such as linear algebra, physics, and computer graphics. It helps in understanding the properties and relationships between vectors and can be used to solve systems of linear equations and determine the basis of a vector space.

3. How do you find the dimension of the span of vectors?

To find the dimension of the span of vectors, you can use various methods such as Gaussian elimination, matrix operations, or finding the rank of a matrix. These methods involve transforming the given vectors into a matrix and performing operations to determine the number of linearly independent vectors.

4. Can the dimension of the span of vectors be greater than the number of given vectors?

Yes, it is possible for the dimension of the span of vectors to be greater than the number of given vectors. This happens when the given vectors are linearly dependent, meaning that one or more vectors can be expressed as a linear combination of the others. In this case, the dimension of the span is equal to the number of linearly independent vectors, which can be less than or equal to the total number of given vectors.

5. How does finding the dimension of the span of vectors relate to vector spaces?

The dimension of the span of vectors is directly related to the dimension of the vector space in which the vectors exist. If the dimension of the span is equal to the dimension of the vector space, it means that the given vectors span the entire space. If the dimension of the span is less than the dimension of the vector space, it means that the given vectors span a subspace of the vector space.

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