Find a subset of the given vectors that forms a basis for...

Find a subset of the given vectors that forms a basis for the space spanned by those vectors, and then express each vector that is not in the basis as a linear combination of the basis vectors. $$\begin{aligned} &\mathbf{v}_{1}=(1,0,1,1), \mathbf{v}_{2}=(-3,3,7,1)\\ &\mathbf{v}_{3}=(-1,3,9,3), \mathbf{v}_{4}=(-5,3,5,-1) \end{aligned}$$

Find a subset of the given vectors that forms a basis for the space spanned by those vectors, and then express each vector that is not in the basis as a linear combination of the basis vectors.
$$\begin{aligned}
&\mathbf{v}_{1}=(1,0,1,1), \mathbf{v}_{2}=(-3,3,7,1)\\
&\mathbf{v}_{3}=(-1,3,9,3), \mathbf{v}_{4}=(-5,3,5,-1)
\end{aligned}$$

Key Concepts

Vector Space and Span

In linear algebra, vectors in a space can combine to form new vectors, and the span of a set of vectors is the collection of all possible linear combinations of those vectors. This is a foundational concept because determining the span helps to understand the coverage and structure of the vector space or subspace in question.

Basis

A basis of a vector space or subspace is a set of vectors that are both linearly independent and span the space. It serves as the minimal and most efficient set of vectors needed to represent every element in that space. Finding a basis eliminates redundancy and provides a clear structure for the space.

Linear Independence

A set of vectors is said to be linearly independent if no vector in the set can be expressed as a linear combination of the others. This concept is crucial because if a vector can be written as a combination of others, it can be excluded from the basis without losing the ability to span the space.

Linear Combination

A linear combination of vectors involves multiplying each by a scalar and adding them together. Expressing vectors as linear combinations of a basis emphasizes the role of the basis in reconstructing any vector in the space and provides a way to distribute representations across the basis vectors.

Gaussian Elimination and Row Reduction

Gaussian elimination is a method used to systematically simplify a set of vectors or a matrix to determine dependencies among them. It is commonly used to identify a set of linearly independent vectors (a basis) by reducing redundancy, and hence to represent any remaining vector as a linear combination of the chosen basis vectors.

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