- #1
heytheree said:
A linearly independent subset of a vector space is a collection of vectors that cannot be written as a linear combination of each other. This means that no vector in the subset can be expressed as a scalar multiple of another vector in the subset.
To determine if a subset is linearly independent, you can use the definition of linear independence. You can also use the concept of linear dependence, where a subset is linearly dependent if at least one vector in the subset can be written as a linear combination of the other vectors in the subset.
Linear independence is important in vector spaces because it allows us to easily solve systems of linear equations and perform other operations such as finding bases and spanning sets. It also helps us understand the structure of vector spaces and how vectors interact with each other.
No, a subset of a vector space cannot be both linearly independent and linearly dependent. These two concepts are mutually exclusive. A subset is either linearly independent or linearly dependent, but not both.
The dimension of a vector space is equal to the number of vectors in any linearly independent subset that spans the entire vector space. This means that the dimension of a vector space is the minimum number of vectors needed to form a basis for that vector space.
