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. In other words, find the dimension of the subspace spanned by the vectors, and a basis for it. Write each of the given vectors as a linear combination of the basis vectors. $$ (0,1,1),(-1,5,3),(1,0,2),(2,-15,1) $$

Key Concepts

Linear Combination

A linear combination involves summing scalar multiples of vectors, where each vector is multiplied by a coefficient. Expressing a vector as a linear combination of others is a foundational tool in linear algebra, used to analyze dependencies, to change bases, and to solve systems of linear equations. It helps in demonstrating connections between vectors within a space.

Dimension

Dimension is defined as the number of vectors in any basis of a vector space or subspace, serving as a measure of the space's size or complexity. It indicates how many independent directions exist in the space, and in practical problems, determining the dimension helps in understanding the structure and capacity of the subspace created by a set of vectors.

Span

The span of a set of vectors is defined as the collection of all possible linear combinations that can be formed with those vectors. It represents the smallest subspace that contains all the given vectors. Understanding the span of a set of vectors is important because it tells us what parts of the vector space can be reached or represented using those vectors.

Linear Independence

This concept refers to a set of vectors where no vector in the set can be written as a linear combination of the others. In other words, the only solution to the equation formed by a linear combination of these vectors equaling the zero vector is the trivial solution where all coefficients are zero. Determining linear independence is essential in understanding whether the vectors provide unique directions in the vector space.

Basis

A basis of a vector space or subspace is a set of linearly independent vectors that span the entire space. This concept is fundamental because it provides the simplest description of the space and allows every vector in the space to be uniquely represented as a linear combination of the basis vectors. Finding a basis reduces the vector space to its core components.

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