Page 266 - Advanced engineering mathematics

246 CHAPTER 7 Matrices and Linear Systems

n
m
n
The null space of a linear transformation T : R → R is the set of all vectors in R that
m
n
T maps to the zero vector in R . Thus X in R is in the null space of T exactly when
m
T (X) =< 0,0,··· ,0 > in R .
We can determine this null space from A T . In terms of matrix multiplication, T (X) is com-
puted as A T X, in which X is an n × 1 column matrix. Thus X is in the null space of T exactly
when X is a solution of
A T X = O.
The null space of T is exactly the solution space of the homogeneous linear system A T X=O.
This solution space is a subspace of R and, because A T has n columns, it has dimension
n
n −rank(A T ). This proves the following.

THEOREM 7.21

n
n
m
Let T : R → R be a linear transformation. Then the null space of T is a subspace of R of
dimension n − rank(A T ).
The dimension of the null space of T is also n minus the number of nonzero rows in the
reduced form of A T .
Algebraists often refer to the null space of a linear transformation as its kernel.
n
m
We have seen that every linear transformation from R to R has a matrix representation. In
the other direction, every m ×n matrix A of real numbers is the matrix of a linear transformation,
defined by T (X) = Y if AX = Y. In this sense linear transformations and matrices are equivalent
bodies of information. However, matrices are better suited to computation, particularly using
software packages. For example, the rank of the matrix of a linear transformation, which we can
find quickly using MAPLE, tells us the dimension of the transformation’s null space.
As a final note, observe that a linear transformation actually has many different matrix rep-
n
m
resentations. We defined A T in the most convenient way, using standard bases for R and R .
If we used other bases, we could still write matrix representations, but then we would have to
use coordinates of vectors with respect to these bases, and these coordinates might not be as
convenient to compute.

SECTION 7.10 PROBLEMS

In each of Problems 1 through 10, determine whether or 4. T (x, y, z,v,w) =<w,v, x − y, x − z,w − x − 3y >
not the given function is a linear transformation. If it is, 5. T (x, y, z,u,v) =< x − u, y − z,u + v>
write the matrix representation of T (using the standard
bases) and determine if T is onto and if T is one-to-one. 6. T (x, y, z,u) =< x + y + 4z − 8u, y − z − x >
Also determine the null space of T and its dimension. 7. T (x, y) =< x − y,sin(x − y)>
1. T (x, y, z) =< 3x, x − y,2z > 8. T (x, y,w) =< 4y − 2x, y + 3x,0,0 >
2. T (x, y, z,w) =< x − y, z − w> 9. T (x, y,u,v,w) =< u − v − w,w + u, z,0,1 >
3. T (x, y) =< x − y, x + y,2xy,2y, x − 2y > 10. T (x, y, z,v) =< 3z + 8v − y, y − 4v>

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October 14, 2010 14:23 THM/NEIL Page-246 27410_07_ch07_p187-246