602
edits
Linear combinations, span, and basis vectors: Difference between revisions
Jump to navigation
Jump to search
Linear combinations, span, and basis vectors (view source)
Revision as of 08:06, 7 November 2021
, 7 November 2021→How do we know if a vector is linearly dependent?
No edit summary |
|||
| (8 intermediate revisions by the same user not shown) | |||
| Line 12: | Line 12: | ||
A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space. | A vector space can have several bases; however all the bases have the same number of elements, called the dimension of the vector space. | ||
====How do we know if a vector is linearly dependent?==== | |||
{{#ev:youtube|https://www.youtube.com/watch?v=k7RM-ot2NWY|250|left|||start= 495&end= 545&loop=1}} | |||
Given a set of vectors, we can determine if they are linearly independent by writing the vectors as the columns of the matrix A, and solving Ax = 0. If there are any non-zero solutions, then the vectors are linearly dependent. If the only solution is x = 0, then they are linearly independent. | Given a set of vectors, we can determine if they are linearly independent by writing the vectors as the columns of the matrix A, and solving Ax = 0. If there are any non-zero solutions, then the vectors are linearly dependent. If the only solution is x = 0, then they are linearly independent. | ||
In other words, if there is a nontrivial linear combination of the vectors that equals the zero vector. If no such linear combination exists, then the vectors are said to be linearly independent. | |||
span | span | ||