Find a basis for the subspace of R^4 that is spanned by the...

Find a basis for the subspace of $R^{4}$ that is spanned by the vectors $$\begin{array}{l} \mathbf{v}_{1}=(1,1,1,1), \quad \mathbf{v}_{2}=(2,2,2,0), \quad \mathbf{v}_{3}=(0,0,0,3) \\ \mathbf{v}_{4}=(3,3,3,4) \end{array}$$

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Key Concepts

Row Reduction

Row reduction, often performed through Gaussian elimination, is a technique used to simplify a set of vectors or a system of equations. This method helps in determining linear independence and in removing redundant vectors, which is essential when finding a basis for a subspace.

Basis

A basis for a vector subspace is a minimal set of linearly independent vectors that spans the entire subspace. Finding a basis involves selecting the smallest number of vectors that still allows every vector in the subspace to be expressed as a unique linear combination of these basis vectors.

Linear Independence

A set of vectors is said to be linearly independent if none of the vectors in the set can be expressed as a combination of the others. This concept is crucial for identifying a basis, as a basis must consist of vectors that are all linearly independent, ensuring that the representation of any vector in the subspace is unique.

Span

The span of a set of vectors is the collection of all linear combinations of those vectors. It defines the subspace that is generated by the given vectors and is fundamental in understanding how vectors can combine to cover a region in the vector space.

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