In Exercises 15-18, find a basis for the space spanned by...

In Exercises $15-18,$ find a basis for the space spanned by the given vectors, $\mathbf{v}_{1}, \ldots, \mathbf{v}_{5}$ $$ \left[\begin{array}{l}{1} \\ {0} \\ {0} \\ {1}\end{array}\right],\left[\begin{array}{r}{-2} \\ {1} \\ {-1} \\ {1}\end{array}\right],\left[\begin{array}{r}{6} \\ {-1} \\ {2} \\ {-1}\end{array}\right],\left[\begin{array}{r}{5} \\ {-3} \\ {3} \\ {-4}\end{array}\right],\left[\begin{array}{r}{0} \\ {3} \\ {-1} \\ {1}\end{array}\right] $$

In Exercises $15-18,$ find a basis for the space spanned by the given vectors, $\mathbf{v}_{1}, \ldots, \mathbf{v}_{5}$
$$
\left[\begin{array}{l}{1} \\ {0} \\ {0} \\ {1}\end{array}\right],\left[\begin{array}{r}{-2} \\ {1} \\ {-1} \\ {1}\end{array}\right],\left[\begin{array}{r}{6} \\ {-1} \\ {2} \\ {-1}\end{array}\right],\left[\begin{array}{r}{5} \\ {-3} \\ {3} \\ {-4}\end{array}\right],\left[\begin{array}{r}{0} \\ {3} \\ {-1} \\ {1}\end{array}\right]
$$

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Key Concepts

Row Reduction

Row reduction, or Gaussian elimination, is a method used to simplify matrices to row echelon form or reduced row echelon form. This process helps identify pivot positions, leading to the determination of linear independence among vectors and the identification of a basis for the span of a given set of vectors.

Linear Independence

Linear independence refers to a property of a set of vectors where no vector can be expressed as a linear combination of the others. This concept is essential for identifying whether a set of vectors can serve as a basis, as a basis must consist of linearly independent vectors.

Basis

A basis of a vector space (or subspace) is a set of linearly independent vectors that span the space. It provides the minimal set of vectors needed to represent every element of the space uniquely. Understanding bases is crucial for simplifying complex vector spaces and for performing coordinate transformations.

Span

The span of a set of vectors is the set of all possible linear combinations of those vectors. It represents the entire subspace that can be generated by the vectors. This concept is fundamental in understanding how vectors can create spaces and how various subspaces relate to each other in linear algebra.

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