Find the the nullity of T and give a geometric description of the kernel and Range of T fo rthe following linear transformation:
T is the projectiononto the xy- coordinate plane:
T(x,y,z) = (x,y,0)

Question

Find the the nullity of T and give a geometric description of the kernel and Range of T fo rthe following linear transformation:
T is the projectiononto the xy- coordinate plane:
T(x,y,z) = (x,y,0)

phi · Answer

T = [1 0 0
        0 1 0
        0 0 0]
That sound right?  If so, you need to find its null space

anonymous · Answer

amistre what do u think?

amistre64 · Answer

thats a good kernel alright

amistre64 · Answer

well, a kernel creates Ax=0 tho right?

anonymous · Answer

yup

amistre64 · Answer

1 0 0
0 1 0
0 0 0

doesnt augment any but it does have a free variable:


       x1     0           0
x =  x2 = 0 + x3  0
       x3     0           1

to which the kernel i believe is the vectors from this

                     0   0
nullA = span: 0 , 0
                      0   1

anonymous · Answer

so the nullity is the kernel?

amistre64 · Answer

http://www.wolframalpha.com/input/?i=kernel+%7B%7B1%2C0%2C0%7D%2C%7B0%2C1%2C0%7D%2C%7B0%2C0%2C0%7D%7D nullity is a dimension of the null space; in this case i believe its 1

anonymous · Answer

so in this case the nullity is 2

anonymous · Answer

good old wolfram i forgot all abt that

amistre64 · Answer

no, since rankT is 2, and T has 3 columns; nullity is 1

anonymous · Answer

ohhhh ya i see

amistre64 · Answer

i spose the 0,0,0 vector is pointless :)
 
             0
nullA =  0
             1

anonymous · Answer

ok then it makes sense

amistre64 · Answer

and that makes a little sense since the null space is orthogonal to the column space.

the colA is the span that creates the surface for xy; and the zaxis itself is the otrhogonal vector to it

anonymous · Answer

okk got it thanks amistre