[METHODS] Coordinates and bases

Reduce a spanning set to a basis

For a finite set $S=\{u_1,\ldots,u_n\}\subseteq\mathbb R^m$ :

Form the matrix $A=[\,u_1\ \cdots\ u_n\,]$ .
Row reduce $A$ to echelon form.
Keep the original vectors whose columns correspond to pivot columns of the echelon form.

The remaining original vectors form a basis for $\operatorname{span}(S)$ .

Row operations identify which columns are redundant, but the basis must use the corresponding columns of the original matrix. Removing the redundant vectors preserves the span.

Extend an independent set to a basis

For an independent set $S=\{u_1,\ldots,u_n\}\subseteq\mathbb R^m$ :

Form the matrix with columns

[\,u_1\ \cdots\ u_n\ e_1\ \cdots\ e_m\,],

where $e_1,\ldots,e_m$ are the standard basis vectors. 2. Row reduce this matrix to echelon form. 3. Keep the original columns corresponding to all pivot columns.

The resulting vectors form a basis for $\mathbb R^m$ containing $S$ .

The standard basis supplies candidate vectors outside the current span. The pivot columns keep the given independent vectors and add only the candidates needed to reach dimension $m$ . As before, take the columns from the original matrix: row reduction changes the vectors but not the pivot positions.

Compute a coordinate vector

To find $[u]_B$ for an ordered basis $B=(u_1,\ldots,u_n)$ :

Solve

u=x_1u_1+\cdots+x_nu_n

for the coefficients. 2. Place the coefficients in the order determined by $B$ :

[u]_B= \begin{pmatrix} x_1\\ \vdots\\ x_n \end{pmatrix}.

Step 1 is a span membership problem and its solution is unique because $B$ is independent.

Methods from Coordinates and bases

Reduce a spanning set to a basis

Extend an independent set to a basis

Compute a coordinate vector

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