Page 157 - A Course in Linear Algebra with Applications

5.3: Operations with Subspaces 141

and denote by U and W the subspaces of R 4 generated by
columns 1 and 2, and by columns 3 and 4 of M respectively.
Find a basis for U + W.
Apply the procedure for finding a basis of the column
space of M. Putting M in reduced column echelon form, we
obtain
/ l 0 0 0\
0 1 0 0
0 0 1 0 '
\ 3 - 1 / 3 - 2 / 3 0 /
The first three columns of this matrix form a basis of U + W;
hence dim(C7 + W)=3.

Example 5.3.6
Find a basis of U fl W where U and W are the subspaces of
Example 5.3.5.
Following the procedure indicated above, we put the ma-
trix M in reduced row echelon form:

/ l 0 0 - 1 \
0 1 0 - 1 j
0 0 1 1 I '
\ 0 0 0 0 /

From this a basis for the null space of M can be read off, as
described in the paragraph preceding 5.1.7; in this case the
basis has the single element

1
- 1 "
V i/

Therefore a basis for U D W is obtained by taking the linear
combination of the generating vectors of U corresponding to