I have a problem understanding getting the KERNEL and IMAGE of a linear transformation. We have the

doodverft05

doodverft05

Answered question

2022-06-06

I have a problem understanding getting the KERNEL and IMAGE of a linear transformation. We have the following transformation given:
R 2 [ x ] R 2 [ x ]
( ϕ ( p ) ) ( x ) = ( x p ( x + 1 ) ) 2 p ( x )
We first have to find its matrix in basis
{ 1 , x , x 2 }
which I know how to get. The transformation matrix result is:
[ 1 1 1 0 0 4 0 0 1 ]
How do I get the KERNEL and the IMAGE from it ?

Answer & Explanation

Dustin Durham

Dustin Durham

Beginner2022-06-07Added 31 answers

The image is generated by vectors that have the columns of the matrix as coordinates. Therefore the image is generated by 1 and 1 + 4 x + x 2 . These vectors are also linearly independent, so they form a basis. The kernel is the set of vectors whose coordinates X solve M X = 0, where M is the matrix. Solving the system you get that the kernel is formed by vectors with coordinates ( a , a , 0 ), thus a basis for the Kernel is the vector 1 + x.
Ayanna Trujillo

Ayanna Trujillo

Beginner2022-06-08Added 13 answers

To find the kernel just solve the homogeneous system
A X = 0 , X = [ x 1 , x 2 , x 3 ] T .
The image is the space spanned by the column vectors of A.

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