In each part, find a basis for the given subspace of R^4,...

In each part, find a basis for the given subspace of $R^{4},$ and state its dimension.
(a) All vectors of the form $(a, b, c, 0)$
(b) All vectors of the form $(a, b, c, d),$ where $d=a+b$ and $c=a-b$
(c) All vectors of the form $(a, b, c, d),$ where $a=b=c=d$

Key Concepts

Spanning Set

A spanning set of a vector space is a set of vectors such that any vector in the space can be expressed as a linear combination of these vectors. Identifying a spanning set is key to demonstrating that a collection of vectors can cover the entire space, an essential step in forming a basis.

Parameterization

Parameterization involves expressing the vectors of a subspace in terms of free variables when the subspace is defined by one or more linear constraints. This technique simplifies the process of finding a basis by clearly showing the independent parameters that capture the full structure of the subspace.

Linear Independence

Linear independence is a property of a set of vectors where no vector in the set can be written as a linear combination of the others. This concept is critical in identifying a basis because a basis must consist of linearly independent vectors to ensure a unique representation for every vector in the space.

Dimension

The dimension of a vector space or subspace is defined as the number of vectors in any basis of the space. It quantitatively describes the degree of freedom within the space and indicates the minimum number of coordinates required to represent any vector in that space.

Basis

A basis of a vector space (or subspace) is a set of vectors that is both linearly independent and spans the entire space. This set serves as the minimum collection of vectors needed to express every element of the space through a unique linear combination, representing the fundamental directions in the space.

Subspace

A subspace is a subset of a vector space that is itself a vector space under the same addition and scalar multiplication operations. It must include the zero vector and be closed under both vector addition and scalar multiplication. Subspaces provide structured environments in which many properties of vector spaces can be studied in a more constrained setting.

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