Page 142 - A Course in Linear Algebra with Applications

126 Chapter Five: Basis and Dimension

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6. If in the last problem dim(5 ) = dim(V), show that S = V.
7. If V is a vector space of dimension n, show that for each
integer i satisfying 0 < i < n there is a subspace of V which
has dimension i.

( 6 \
8. Write the transaction I —4 as a linear combination of
v-v

simple transactions.
9. Prove that vectors A, B, C generate R 3 if and only if
none of these vectors belongs to the subspace generated by
the other two. Interpret this result geometrically.
10. If V is a vector space with dimension n over the field of
two elements, prove that V contains exactly 2 n vectors.

5.2 The Row and Column Spaces of a Matrix

Let A be an m x n matrix over some field of scalars F.
Then the columns of A are m-column vectors, so they belong
m
to the vector space F , while the rows of A are n-row vectors
and belong to the vector space F n. Thus there are two natural
subspaces associated with A, the row space, which is generated
by the rows of A and is a subspace of F n, and the column space,
m
generated by the columns of A, which is a subspace of F .
We begin the study of these important subspaces by in-
vestigating the effect upon them of applying row and column
operations to the matrix.

Theorem 5.2.1
Let A be any matrix.
(i) The row space is unchanged when an elementary row
operation is applied to A.
(ii) The column space is unchanged when an elementary
column operation is applied to A.