Page 147 - A Course in Linear Algebra with Applications

5.2: The Row and Column Spaces of a Matrix 131

ciwi + c 2 w 2 -I 1- c fcw fc is surely c xX x + c 2X 2 H h CkX k.
Hence the set of all coordinate column vectors of elements of S
equals the subspace T of F n which is generated by X x,..., X k •
Moreover wi, W2,. • •, w/. will be linearly independent if and
only if Xi, X 2 ,..., X k are. In short wi, w 2 ,..., w*; form a
basis of S if and only if X\, X 2, •. •, X k form a basis of T; thus
our problem is solved.

Example 5.2.2
Find a basis for the subspace of Pt(R) generated by the poly-
2
3
3
3
nomials 1 — x — 2x , 1 + x , 1 + x + 4x , x .
Of course we will use the standard ordered basis for P^TV)
2
3
consisting of l,x,x ,x . The first step is to write down the
coordinate vectors of the given polynomials with respect to the
standard basis and arrange them as the columns of a matrix
A; thus
/ 1 1 1 0 \
- 1 0 1 0
0 0 0 1 '
V-2 1 4 0/

To find a basis for the column space of A, use column opera-
tions to put it in reduced column echelon form:

/ l 0 0 0 \
0 1 0 0
0 0 1 0 '
\ 1 3 0 0 /

The first three columns form a basis for the column space
of A. Therefore we get a basis for the subspace of P4(R) gen-
erated by the given polynomials by simply writing down the
polynomials that have these columns as their coordinate col-
umn vectors; in this way we arrive at the basis

3 3 2
l + x , x + 3x , x .