Example 1: Image, rank, nullspace and nullity
Consider the linear mapping A : R3 -> R2, A( x1, x2, x3 ) = ( x1, x1 + x2 ), i.e., the matrix
A = 1 0 0
1 1 0
Find the image, the rank, the nullspace and the nullity of A.

Solution:

R(A) = { = ( y1, y2 ) | there exists an = ( x1, x2, x3 ) : A = }

= A <=> y1 =
y2
1 0 0
1 1 0
x1 =
x2
x3
x1
x1 + x2
<=> y1 = x1
y2 = x1 + x2
<=> x1 = y1 (x3 arbitrary)
x2 = y2 - y1
=>R(A) = R2 (the image ofA)
=>dim R(A) = 2 (the rank of A)

Nullspace: A =

<=> 1 0 0
1 1 0
x1
x2
x3
= 0
0
<=> x1
x1 + x2
= 0
0
<=> x1 = 0
x1 + x2 = 0
=> x1 = 0
x2 = 0
x3 arbitrary, denote x3 = c R
 <=> = ( 0, 0, c ) = c ( 0, 0, 1 ), c R

N(A) = { | = c ( 0, 0, 1 ), c R} ( the nullspace of A; a basis for N(A) is {( 0, 0, 1 )} )

dim N(A) = 1 (the nullity of A)

Note. dim N(A) + dim R(A) = 1 + 2 = 3 = n = the number of columns of A

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