# Linearity / Matrix functions?

• Feb 3rd 2009, 06:06 PM
Unenlightened
Linearity / Matrix functions?
I shouldn't have left it so late to ask this, but the answers to any of these would be most helpful...

Is the function mapping a matrix A to the determinant of A linear?

I'm thinking it's non-linear, but simply because I can't think of a transformation matrix that could map [[a,b],[c,d]] <--- (A 2x2 matrix a, b, c, d reading left to right... apologies for absence of Latex!) to ad-bc...

Is the operation mapping f to f''+3f' linear?
(Where f is over the collection of all infinitely differentiable functions)
I know differentiation is linear, but does it hold for two separate derivatives added together?
Fine, A(cx + dy) = c(Ax)+d(Ay) if and only if the function is linear, but how does one actually go about picking these x and y s, or creating a matrix A to test one's theory?

Is a function mapping f to its second derivative linear?
Aye, I'm presuming this is linear, since differentiation is linear (although I'm not sure how to explain that...)

Is the function mapping a matrix A to its trace linear?
This one is linear, right? Because you're just adding two entries together. Again, I'm not sure how to construct the function as a matrix...

Is the function mapping x to 3x + 2 linear?
This is a straight out 'no', right? 3x is fine, but you can't just add a constant like that, right?

Let C = [[1,2],[3,4]]. Is the function mapping A to AC-CA linear?
Again, I'm guessing it's not, for similar reasons to the previous...

Any help on any of these much appreciated.
• Feb 3rd 2009, 06:32 PM
Isomorphism
Quote:

Originally Posted by Unenlightened
I shouldn't have left it so late to ask this, but the answers to any of these would be most helpful...

Is the function mapping a matrix A to the determinant of A linear?

Idea: What is
$\displaystyle \det( 3A )$?

Quote:

Is the operation mapping f to f''+3f' linear?
(Where f is over the collection of all infinitely differentiable functions)
I know differentiation is linear, but does it hold for two separate derivatives added together?
Fine, A(cx + dy) = c(Ax)+d(Ay) if and only if the function is linear, but how does one actually go about picking these x and y s, or creating a matrix A to test one's theory?

Quote:

Is a function mapping f to its second derivative linear?
Aye, I'm presuming this is linear, since differentiation is linear (although I'm not sure how to explain that...)
For both these questions, you dont need matrices... Check if f and g are mapped as above, is f+g mapped like above....

Quote:

Is the function mapping a matrix A to its trace linear?
This one is linear, right? Because you're just adding two entries together. Again, I'm not sure how to construct the function as a matrix...
Yes it is. Rigorously you say: $\displaystyle \text{Tr }(\alpha A + \beta B) = \alpha \text{Tr } (A) + \beta \text{Tr }(B)$

Quote:

Is the function mapping x to 3x + 2 linear?
This is a straight out 'no', right? 3x is fine, but you can't just add a constant like that, right?
Yes. Show that 0 does not map to 0, thus it is not linear

Quote:

Let C = [[1,2],[3,4]]. Is the function mapping A to AC-CA linear?
Again, I'm guessing it's not, for similar reasons to the previous...

This question is not clear :(

• Feb 3rd 2009, 07:10 PM
Unenlightened
Thankee koindly :)

Sorry about the last one - it's supposed to be the matrix
(1 2)
(3 4)
And the function is to map A onto A*C - C*A...

Ooh also

How about the function mapping A to A transpose?
Non-linear also?
• Feb 4th 2009, 10:14 PM
Isomorphism
Quote:

Originally Posted by Unenlightened
Thankee koindly :)

Sorry about the last one - it's supposed to be the matrix
(1 2)
(3 4)
And the function is to map A onto A*C - C*A...

Ooh also

How about the function mapping A to A transpose?
Non-linear also?

They all are linear. Just apply the definition of linearity to get the answer :)