Find out whether the given vectors are dependent or...

Find out whether the given vectors are dependent or independent; if they are dependent, find a linearly independent subset. Write each of the given vectors as a linear combination of the independent vectors.
$$(0,1,1),(-1,5,3),(1,0,2),(2,-15,1)$$

Key Concepts

Expressing Vectors as Linear Combinations

Expressing vectors as linear combinations involves writing a vector as a sum of scalar multiples of other vectors. This process is foundational in linear algebra because it not only helps in understanding the structure of vector spaces but also provides practical methods for solving systems of linear equations, analyzing the span and basis, and simplifying problems involving vector relationships.

Basis and Spanning Set

A basis of a vector space is a set of linearly independent vectors that spans the entire space. This means every vector in the space can be expressed as a linear combination of the basis vectors. The concept of a basis is central to linear algebra as it provides a framework for understanding vector spaces, determining dimensions, and simplifying computations.

Linear Independence

Linear independence is a property of a set of vectors in which no vector in the set can be represented as a linear combination of the others. This means that the only solution to the equation where a linear combination of these vectors equals the zero vector is when all the coefficients are zero. This concept is crucial for understanding the structure of vector spaces because it provides a way to identify the minimal sets of vectors needed to represent all vectors in the space.

Linear Dependence

Linear dependence occurs when at least one vector in the set can be expressed as a linear combination of the other vectors. This indicates redundancy in the set, implying that some vectors do not contribute new directions or dimensions to the space. Identifying linear dependence is essential in simplifying problems and selecting a basis that efficiently spans the vector space.

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