Range and Null Space - Stanford University
Range and Null Space
Stephen Boyd and Sanjay Lall EE263
Stanford University
1
Nullspace of a matrix
2 the nullspace of A Rm?n is defined as
f 2 j g R null(A) = x
n Ax = 0
I null(A) is set of vectors mapped to zero by y = Ax I null(A) is set of vectors orthogonal to all rows of A
null(A) gives ambiguity in x given y = Ax:
I 2 if y = Ax and z null(A), then y = A(x + z) I 2 conversely, if y = Ax and y = Ax~, then x~ = x + z for some z null(A) N null(A) is also written (A)
2
Zero nullspace A is called one-to-one if 0 is the only element of its nullspace
f g null(A) = 0
Equivalently,
I x can always be uniquely determined from y = Ax
(i.e., the linear transformation y = Ax doesn't `lose' information)
I mapping from x to Ax is one-to-one: different x's map to different y's
I columns of A are independent (hence, a basis for their span)
I 2 R A has a left inverse, i.e., there is a matrix B ?n m s.t. BA = I
IT AA
is
invertible
3
Zero nullspace
f g I if A has a left inverse then null(A) = 0 (proof by contradiction)
I null(A) = null(ATA)
I f g if null(A) =
0
then
A
is
left
invertible,
because
T AA
is
invertible,
so
B
=
(AT A)
1T
A
is
a
left
inverse
4
Two interpretations of nullspace
2 suppose z null(A), and y = Ax represents measurement of x
I z is undetectable from sensors -- get zero sensor readings I x and x + z are indistinguishable from sensors: Ax = A(x + z)
null(A) characterizes ambiguity in x from measurement y = Ax
alternatively, if y = Ax represents output resulting from input x
I z is an input with no result I x and x + z have same result
null(A) characterizes freedom of input choice for given result
5
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