Basis and Dimension of Subspaces

A basis provides the minimal, unique coordinate system needed to span a subspace without redundancy.

Defining a Basis

A set of vectors forms a basis for a vector space if they are linearly independent and span the entire space. This means every vector in the space can be uniquely written as a linear combination of the basis vectors. If even one vector is removed, the set no longer spans the space.

V = \text{span}(\{v_1, v_2, \dots, v_n\})

Dimension of a Space

The dimension of a vector space is defined as the number of vectors in any of its bases. While a single vector space has infinitely many possible bases, every basis for that space always contains the exact same number of vectors. For instance, any basis for three-dimensional space must consist of exactly three vectors.

\dim(V) = n

Finding Basis Vectors

To find a basis for a subspace like the column space of a matrix, reduce the matrix to row echelon form. The pivot columns in the original matrix form a basis for that column space. The number of pivot columns directly gives the rank and dimension of the subspace.

\dim(\text{Col}(A)) = \text{rank}(A)

Flashcard

\mathcal{B} = \{v_1, \dots, v_k\}

A basis set for a vector space

A set of linearly independent vectors that spans the target vector space, allowing every vector in that space to be represented uniquely.

Check your understanding

Suppose a subspace S in four-dimensional space is spanned by four vectors, but row reduction reveals only three pivot columns. What is the dimension of S?

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