Sorry to take up everyone's time... But I know most people here are mathematically inclined and as far as I've seen, very very helpful.. Tomorrow I have this algebra exam and while I think I got the hang of most of it, this (sort of a) problem seems to remain elusive (for me):
Let f be an endomorphism of vector space R R^4 given by
(f)e= this matrix with the lines
(0 1 2 3)
(-1 2 1 0)
(3 0 -1 -2)
(5 -3 -1 1)
in canonical basis.
Determine the dimension and give a basis for each of the following:
Im f; Ker f; Im f+Ker f; Im f (intersected with) Ker f.
I have some of the notes but only for the Ker f thingie, which is quite trivial from the definition that the kernel is every vector x for which f(x)=0 vector. But from there I don't really understand... Im f is every possible result of the linear transformation? But then Imf+Kerf is basically Imf? Or what?
Can anyone pls help me? Thanks very very much!
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basic linear algebra
#1
Posted 2007-January-30, 10:40
... and I can prove it with my usual, flawless logic.
George Carlin
George Carlin
#2
Posted 2007-January-30, 12:15
Do you have a textbook for this? I think you should look that up for the followed notation.
Anyway, Im f, if stands for image is the set of vectors y, such that there is a vector x for which f(x) = y.
The usual definition for A+B i have seen is: if A and B are two vector spaces, then A + B = { a + b | a \in A, b \in B }
It could also be Union of Im f and Ker f, but in that case it might not make much sense to talk about a basis, as it might not be a vector space.
Hope that helps.
Good luck with your exam.
Anyway, Im f, if stands for image is the set of vectors y, such that there is a vector x for which f(x) = y.
The usual definition for A+B i have seen is: if A and B are two vector spaces, then A + B = { a + b | a \in A, b \in B }
It could also be Union of Im f and Ker f, but in that case it might not make much sense to talk about a basis, as it might not be a vector space.
Hope that helps.
Good luck with your exam.
#3
Posted 2007-January-30, 12:38
In this example, a basis for Im f is
(Of course, there are lots of other ways of writing this. Choosing any three columns of the original matrix would work, but it's better to get a more reduced form.)
And a basis for Ker f is
I assume you know how to work out these things.
Im f + Ker f is then the space spanned by all four vectors. Since the four vectors are clearly linearly independent, Im f + Ker f has dimension 4, ie. it's the whole of R^4.
Im f (intersected with) Ker f is {0} (once you have a nice basis for Im f this is easy to see).
If the four vectors weren't linearly independent then Im f + Ker f would have dimension 3; but it would have to contain all of Im f, so it would have to be exactly the same as Im f. In this case, Im f (intersected with) Ker f would be the whole of Ker f.
[Hope the working is correct...]
( 1 ) ( 0 ) ( 0 ) ( 0 ) ( 1 ) ( 0 ) ( 0 ) , ( 0 ) , ( 1 ) ( 1 ) (-2 ) ( 1 )
(Of course, there are lots of other ways of writing this. Choosing any three columns of the original matrix would work, but it's better to get a more reduced form.)
And a basis for Ker f is
( 0 ) ( 1 ) (-2 ) ( 1 )
I assume you know how to work out these things.
Im f + Ker f is then the space spanned by all four vectors. Since the four vectors are clearly linearly independent, Im f + Ker f has dimension 4, ie. it's the whole of R^4.
Im f (intersected with) Ker f is {0} (once you have a nice basis for Im f this is easy to see).
If the four vectors weren't linearly independent then Im f + Ker f would have dimension 3; but it would have to contain all of Im f, so it would have to be exactly the same as Im f. In this case, Im f (intersected with) Ker f would be the whole of Ker f.
[Hope the working is correct...]
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