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Matrix Further Maths

I got the rank as 3 but don't know how to proceed.

Screenshot from 2019-02-27 10-40-19.png

Thanks for help :smile:.
Reply 1
Avatar for mqb2766
mqb2766
21
How did you go about the rank 3 proof?
It's fairly clear that the unity vector is mapped to zero as the rows sum to zero - so this is a basis of the null space of A as it's 1D
Reply 2
Avatar for esrever
esrever
OP
20
Original post by mqb2766
How did you go about the rank 3 proof?
It's fairly clear that the unity vector is mapped to zero as the rows sum to zero - so this is a basis of the null space of A as it's 1D


I reduced the matrix to echelon form.

Can you please explain what you mean by the unity vector being mapped to 0.
Reply 3
Avatar for mqb2766
mqb2766
21
When x = (1,1,1,1), then Ax = 0.
So you could split the x in the question up into the two parts and show that the second part is zero, but thatt would be more verifying than showing.

Edit - depending on the number of marks, you know that X is made from the columns so the lambda part must give a column sum of zero, so perhaps this is what they want.
(edited 5 years ago)
Reply 4
Avatar for esrever
esrever
OP
20
Original post by mqb2766
When x = (1,1,1,1), then Ax = 0.
So you could split the x in the question up into the two parts and show that the second part is zero, but thatt would be more verifying than showing.

Edit - depending on the number of marks, you know that X is made from the columns so the lambda part must give a column sum of zero, so perhaps this is what they want.


Thank you for help :smile:. I am still having a bit of difficulty in understanding because I don't think I understand null space well enough. I will try this question again after reading more about null spaces.
Reply 5
Avatar for esrever
esrever
OP
20
Original post by mqb2766
When x = (1,1,1,1), then Ax = 0.
So you could split the x in the question up into the two parts and show that the second part is zero, but thatt would be more verifying than showing.

Edit - depending on the number of marks, you know that X is made from the columns so the lambda part must give a column sum of zero, so perhaps this is what they want.


regarding the edited part: Do you mean I should just multiply out the matrix and compare left hand side to the right hand side?
Reply 6
Avatar for mqb2766
mqb2766
21
The null space is simply the part of X space which gets mapped to zero by A.
In this case its fairly easy to see that as the matrix is rank 3 and the columns sum to zero, the null space is lambda*(1,1,1,1) and if this is added to any vector x, Ax is unchanged.
Reply 7
Avatar for esrever
esrever
OP
20
Original post by mqb2766
The null space is simply the part of X space which gets mapped to zero by A.
In this case its fairly easy to see that as the matrix is rank 3 and the columns sum to zero, the null space is lambda*(1,1,1,1) and if this is added to any vector x, Ax is unchanged.


Thank you so much :smile:

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