Page 148 - A Course in Linear Algebra with Applications

132 Chapter Five: Basis and Dimension

Hence the subspace generated by the given polynomials has
dimension 3.

Exercises 5.2

1. Find bases for the row and column spaces of the following
matrices:

<-» C2 =S i)-o» ("J J | J)-

2. Find bases for the subspaces generated by the given vectors
in the vector spaces indicated:
2
3
3
2
l
(a) - 2 z - : r , 3x-x , l + x + x +x , 4 + 7x + x 2 + 2x 3
in P 4 (R);

3. Let A be a matrix and let N, R and C be the null space,
row space and column space of A respectively. Prove that

dim( JR) + dim(iV) = dim(C) + dim(iV) = n

where n is the number of columns of A.
4. If A is any matrix, show that A and A T have the same
rank.
5. Suppose that A is an m x n matrix with rank r. What is
T
the dimension of the null space of A 7
6. Let A and B be m x n and n x p matrices respectively.
Prove that the row space of AB is contained in the row space
of B, and the column space of AB is contained in the the
column space of A. What can one conclude about the ranks
of AB and BA ?
7. The rank of a matrix can be defined as the maximum num-
ber of rows in an invertible submatrix: justify this statement.