Question

Find the dimension of and the basis of the kernel: L(x)=(x2, x3)^T which is a linear transformation from R^3 to R^2.

Answer 1

@amistre64 do you remember anything about the above^

Answer 2

the words all sound familiar.

isnt a kernel related to the nulspace?

Answer 3

I'm not entirely sure. I'm looking for information outside my textbook. I'm completely at a loss at this point.

Answer 4

http://www.math.purdue.edu/~beranger/262/ch5/5-3.pdf

Answer 5

That's what I'm looking at now.

Answer 6

the dimension of the kernal, and the basis of the kernel ... so we need to determine 2 of its properties

Answer 7

yes

Answer 8

So the transformation is a coefficient matrix = (1,1)^T ... Then the rref = (1,0)^T . So the dimension is 1 ? I think

Answer 9

and the basis is (1,0)^T

Answer 10

but that doesn't seem right

Answer 11

ill have more time and a clearer head for this later on ... ill read up and try to refresh my memory in the mean time.

Answer 12

okay!

Answer 13

the kernel is related to:
\[\begin{pmatrix}x^2\\x^3\end{pmatrix}\to ax^2+bx^3=0\]

Answer 14

but yeah, ill chk this later ...

Answer 15

http://www.math.lsa.umich.edu/~hochster/419/ker.im.html

this seems about my level of comprehension today :)

Answer 16

R^3 -> R^2, refers us to a 3col, 2row, it also would seem to address a setup such that L(x,y,z) , 3 elements to express R^3, ... = (x^2,x^3) , 2 elements to express R^2.

http://xmlearning.maths.ed.ac.uk/lecture_notes/linear_maps/kernels_images/kernels_images_3.php

L(x) = (x^2, x^3) seems to represent R -> R^2

Answer 17

is that:\[a.~~L(x)=(x^2,x^3)^T\\OR\\b.~~L(x)=(x_2,x_3)^T\]

Answer 18

if its polynomials (x^2, x^3), which the notation would be P->P i believe, we would be finding:

0x^3+1x^2+0x+0 = 0
1x^3+0x^2+0x+0 = 0



if its just vectors for vectors sake (x_2, x_3), then the notation R->R would make more sense to me, and thus finding:

\[0x_3+1x^2+0x_1 = 0\\
1x_3+0x_2+0x_1 = 0\]