- #1
krozer
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If I create a matrix whose columns are the vectors, and then I row-reduce it and there's a zero row, are the vectors lineraly dependent? why?
Linearly dependent vectors are a set of vectors in which at least one vector can be expressed as a linear combination of the others. This means that one vector can be written as a scalar multiple of another vector, or a combination of multiple vectors.
A set of vectors is linearly dependent if there exists a non-trivial solution to the equation c1v1 + c2v2 + ... + cnvn = 0, where c1, c2, ..., cn are scalars and v1, v2, ..., vn are the vectors in the set. This means that there is more than one way to express the zero vector as a linear combination of the vectors in the set.
Linearly dependent vectors are important in linear algebra because they can be used to create a basis for a vector space. A basis is a set of linearly independent vectors that can be used to represent any vector in the vector space. Additionally, linearly dependent vectors can be used to solve systems of linear equations.
Yes, a set of two vectors can be linearly dependent. For example, the vectors (1, 2, 3) and (2, 4, 6) are linearly dependent because one vector is a scalar multiple of the other (2 * (1, 2, 3) = (2, 4, 6)). This means that one vector can be expressed as a linear combination of the other, making the set linearly dependent.
To make a set of linearly dependent vectors linearly independent, you can remove one or more vectors from the set. This will change the number of vectors in the set and may change the span of the vectors, making them linearly independent. Alternatively, you can also scale the vectors so that they are no longer linearly dependent, but this may also change the span of the vectors.