Page 158 - A Course in Linear Algebra with Applications

142 Chapter Five: Basis and Dimension

the scalars in the first two rows of this vector, that is to say

3
+ 1- 0
w

Thus dim(C7 n W) 1.

Example 5.3.7
Find bases for the sum and intersection of the subspaces U and
W of P4CR) generated by the respective sets of polynomials

3
3
2
3
{l + 2x + x , 1 x } and {x + x 2 - 3x , 2 + 2x - 2x }.
x
The first step is to translate the problem to R 4 by writing
down the coordinate columns of the given polynomials with
respect to the standard ordered basis 1, 3 of P 4 (R).
Arranged as the columns of a matrix, these are

1 0 2
/ !
2 - 1 1 2
A =
0 - 1 1 0
\ 1 0 - 3 - 2

Let U* and W* be the subspaces of R 4 generated by the
coordinate columns of the polynomials that generate U and
W, that is, by columns 1 and 2, and by columns 3 and 4 of
A respectively. Now find bases for U* + W* and U* D W*,
just as in Examples 5.3.5 and 5.3.6. It emerges that U* + W*,
which is just the column space of A, has a basis