Finding Dimension and Basis with a given Spanning Set

Consider a vector space $W$ spanned by a set $S$.

1. Form a matrix with the vectors in the spanning set as the rows.
2. Reduce to a matrix in the row echelon form.
3. The number of non-zero rows is the dimension of the vector space $W$.
4. The vectors corresponding to the non-zero rows form the basis of the vector space $W$.

An Alternative method to Row based Method

The row-based method produces a basis from a spanning set, but may not contain the vectors in the spanning set.

If $R$ is the matrix obtained by row reducing $A$, then the columns of $A$ corresponding to the columns of $R$ containing the pivots(i.e., the leading 1’s or equivalently the columns corresponding to the dependent variables) from a basis for the column space of $A$.

Column Method

Let $W$ be the subspace of $\R^3$ spanned by the set $S$.

1. Form the matrix with the vectors in $S$ as the columns.
2. Row reduce this matrix.
3. Find the columns in which the pivots are found.
4. These columns will correspond to the vectors in $S$, by the matrix formed in 1.
5. These vectors in $S$ form a basis for $W$.