Find dimension and ker of matrices ?

[sid9221]

Find dimension and ker of matrices ??

Let V be an F-vector space and (phi:v->v) be an F-linear transformation of V . Define what
it means for a vector v ε V to be an eigenvector of phi and what is meant by the associated
eigenvalue.

This is the form of the question during my calculations I need to calculate:


Now I have from the eigenvalues:

dim(ker()) of a matrix:
1 4 -3
0 0 0
0 0 0

and

dim(ker()) of a matrix:
1 0 -2/7
0 1 -5/7
0 0 0

Then when the I add up the the dim's I will be able to tell if it is diagonalisable.

Now in this case it is obvious, the dim(ker()) of one is 1 and another is 2. But I can't tell which one is which.

What I mean is the 1st Matrix = 2 and the second Matrix = 1 or viceversa ?

In this case in either way it will add upto 3 and as the original matrix was a 3x3 matrix it is diagonalisable right ?

In essence I don't really understand what dim or ker do ...

 

[HallsofIvy]

I have no clue what you are asking. The title is "Find dimension and ker of matrices" but your first question appears to be about eigenvalues. You have "dim(ker()) of a matrix:" followed by what look like row reduced matrices. What are the original matrices?

You say "Then when the I add up the the dim's I will be able to tell if it is diagonalisable." Tell if what is diagonalisable?

The dimension of the kernel of a matrix tells you whether or not 0 is an eigenvalue and the algebraic dimension but tells you nothing about any non-zero eigenvalues- and does NOT tell you if the matrix is diagonalizable. In particular a row-reduced matrix does not have the same eigenvalues or eigenvectors as the original matrix.

 

[sid9221]

I have no clue what you are asking. The title is "Find dimension and ker of matrices" but your first question appears to be about eigenvalues. You have "dim(ker()) of a matrix:" followed by what look like row reduced matrices. What are the original matrices?

You say "Then when the I add up the the dim's I will be able to tell if it is diagonalisable." Tell if what is diagonalisable?

The dimension of the kernel of a matrix tells you whether or not 0 is an eigenvalue and the algebraic dimension but tells you nothing about any non-zero eigenvalues- and does NOT tell you if the matrix is diagonalizable. In particular a row-reduced matrix does not have the same eigenvalues or eigenvectors as the original matrix.

My bad, I shouldn't have given the question in the first place as I just want to know what the resective dims kers are.

1)
what is the dim(ker()) of:
1 4 -3
0 0 0
0 0 0

2)
and what is the dim(ker()) of:
1 0 -2/7
0 1 -5/7
0 0 0

What are the answers to 1) and 2) is all I want to know, also the dims of the above matrices will also be useful.

I know that the answers are 1 and 2, but I'm not sure which one is 1 and which one is 2.

If I am coming across confused, its because I am...sorry

 

[sachav]

The ranks of both matrices are obvious (just look at the columns). The nullities follow immediately with the rank-nullity theorem.

 

[sid9221]

The ranks of both matrices are obvious (just look at the columns). The nullities follow immediately with the rank-nullity theorem.

So for the first one the dim(ker()) is 3-1=2
and for the second one the dim(ker()) is 3-2=1

where 3 is # of columns and 1 and 2 are the ranks respectively ?

 

[sachav]

Yes, that's correct.