How to know arbitrary vectors in a given matrix?
Understand the Problem
The question is asking for a method or criteria to determine arbitrary vectors within a specified matrix. This relates to understanding linear combinations and the span of vectors in linear algebra.
Answer
Express vectors as linear combinations of column vectors or identify via the column space basis.
You can determine arbitrary vectors in a given matrix by expressing them as linear combinations of column vectors of the matrix or by identifying them through the basis of the column space of the matrix.
Answer for screen readers
You can determine arbitrary vectors in a given matrix by expressing them as linear combinations of column vectors of the matrix or by identifying them through the basis of the column space of the matrix.
More Information
In linear algebra, understanding the span and basis of a matrix's column vectors can help identify potential arbitrary vectors related to that matrix.
Tips
A common mistake is to assume any vector in a matrix can be arbitrary without considering the matrix's basis or rank.
Sources
- Finding the image of an arbitrary vector - math.stackexchange.com
- Find the image of an arbitrary vector | Wyzant Ask An Expert - wyzant.com
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