Math 324: Assignment 6
Instructions. Exercises 1, 2, 3 are to be written and turned in on Monday, October 25.
Exercises 2(a), 4, 5, 6 and 7 are for in-class presentation.
Exercise 1. Consider the matrix
A=
1 0 −2 1 5
3 1 −5 0 8
1 2 0 −5−9
Let the row-space of Abe the subspace of IR5generated by the rows of A. Let the column-space
of Abe the subspace of IR3generated by the columns of A.
(a) Find a basis for the row-space of A.
(b) Express every row in A as a linear combination of basis vectors found in (a).
(c) Find a basis for the column space of A.
(d) Express every column in A a linear combination of the basis vectors found in (c).
(e) What are the dimensions of the row-space and column space? Are they equal to each other?
Will they always be equal to each other, irrespective of the matrix?
(f) Find the null-space of LAand find a basis for the null-space of LA.
(g) For an mby nmatrix, the dimension theorem states that
n= dim(N(A)) + rank(A).
Verify that this is true for the matrix above.
Exercise 2. (a) Use matrix multiplication to show that any solution to Ax =bcan be written
as x=xp+xhwhere Axp=band Axh= 0.
(b) Verify this with the system of equations Ax =bwhere
A=
0 3 −6−4−3
−1 3 −10 −4−4
2−62028
and b=
−5
−2
−8
Exercise 3. Do Exercise 11, p. 181.
Exercise 4. Suppose T:V→Wis a linear transformation, and {v1, v2, . . . , vn}is a basis for
V. Show that {T(v1), T (v2), . . . , T (vn)}generates R(T), the range of T.
