Linear algebra - Image and Kernel

[sincera4565]

Homework Statement

Let V be a 3 dim vector space over F and e_1 e_2 and e_3 be those fix basis
The question provide us with the linear transformation T\in L(V) such that
T(e_1) = e_1 + e_2 - e_3
T(e_2) = e_2 - 3e_3
T(e_3) = -e_1 -3e_2 -2e_3

we are ask to find the matrix of T and the basis of ker(T) and Im(T)

2. The attempt at a solution

I think I find the matrix right
where the matrix of T should be
1 0 -1
1 1 -3
-1 -3 -2

but the problem is I am not sure how can i find the ker(T) and Im(T)

 

[MednataMiza]

Few pointers:
o The matrix you wrote down is wrong. Look for the proper way to order the coordinates of the basis vectors.
o For Im(T): You have to find the span of vectors
o For Ker(T): You need to solve a matrix equation

 

[Deveno]

Few pointers:
o The matrix you wrote down is wrong. Look for the proper way to order the coordinates of the basis vectors.

i'm not seeing this. it looks correct to me.

o For Im(T): You have to find the span of vectors
o For Ker(T): You need to solve a matrix equation

if one uses row-reduction, one could accomplish both at the same time.

 

[MednataMiza]

Am I wrong or one should order the vectors in columns not in rows ?
Finding the span is just row-reducing the matrix ...

 

[Deveno]

Am I wrong or one should order the vectors in columns not in rows ?
Finding the span is just row-reducing the matrix ...

it appears that is what has been done.

T(e₁) = e₁ + e₂ - e₃,

that is: T((1,0,0)^T) = 1(1,0,0)^T + 1(0,1,0)^T + (-1)(0,0,1)^T

= (1,1,-1)^T, which appears to be the first column of sincera4565's matrix.