Linear transformation, subspace and kernel

[Tala.S]

Hi

We have a linear transformation g : ℝ^2x2 → ℝ g has U as kernel,

U: the 2x2 symmetric matrices

(ab)
(bc)

A basis for U is

(10)(01)(00)
(01)(10)(01)I thought this would be easy but I've been sitting with the problem for a while and I have no clue on how to solve it (maybe because I don't fully understand it).

But this is how I understand it : we need to find a linear transformation that transforms symmetric 2x2 matrices (R^4) to 1x1 matrices (R), so we have

g(a,b,b,c) = (a b b c) = 0

?

 

[Dick]

Hi

We have a linear transformation g : ℝ^2x2 → ℝ g has U as kernel,

U: the 2x2 symmetric matrices

(ab)
(bc)

A basis for U is

(10)(01)(00)
(01)(10)(01)


I thought this would be easy but I've been sitting with the problem for a while and I have no clue on how to solve it (maybe because I don't fully understand it).

But this is how I understand it : we need to find a linear transformation that transforms symmetric 2x2 matrices (R^4) to 1x1 matrices (R), so we have

g(a,b,b,c) = (a b b c) = 0

?

How about this? For a general 2x2 matrix [[a,b],[c,d]] you can write a linear transformation to R as g(a,b,c,d)=w*a+x*b+y*c+z*d. You have to find w,x,y,z such that g=0 if and only if the matrix is symmetric.

 

[Tala.S]

But how can we find the values for w,x,y and z so that g = 0 when we don't even know the values of a,b,b and c ?

 

[Dick]

But how can we find the values for w,x,y and z so that g = 0 when we don't even know the values of a,b,b and c ?

Try sample matrices. Like your basis for U.

 

[Tala.S]

like this :

a * [[1,0],[0,0]] + b * [[0,1],[1,0]] + c * [[0,0],[0,1]] = 0 ?

 

[Dick]

like this :

a * [[1,0],[0,0]] + b * [[0,1],[1,0]] + c * [[0,0],[0,1]] = 0 ?

No. Take a matrix in your basis. Like [[1,0],[0,0]]. Figure out what a,b,c and d are. Then put them into the form for g. You want g to give you a real number, not a matrix.

 

[Tala.S]

a * [[1,0],[0,0]] + b * [[1,0],[0,0]] + c * [[1,0],[0,0]] +d* [[1,0],[0,0]] = 0

(a,b,c,d) = 0

But shouldn't it be a [[a,b],[b,c]] matrix ?

 

[Dick]

a * [[1,0],[0,0]] + b * [[1,0],[0,0]] + c * [[1,0],[0,0]] +d* [[1,0],[0,0]] = 0

(a,b,c,d) = 0

But shouldn't it be a [[a,b],[b,c]] matrix ?

No again. Let's back up. Let's take a basis of R2x2. e1=[[1,0],[0,0]], e2=[[0,1],[0,0]], e3=[[0,0],[1,0]], e4=[[0,0],[0,1]]. That's a basis for R2x2, right? What should g(e1) be?

 

[Tala.S]

g(e1) = [[1,0,0,0]]

g(e2) = [[0,1,0,0]]

g(e3) = [[0,0,1,0]]

g(e4) = [[0,0,0,1]]

 

[Dick]

g(e1) = [[1,0],[0,0]]

g(e2) = [[0,1],[1,0]]

g(e3) = [[0,0],[0,1]]

No no. g is a linear transformation from R2x2 to R! R is the real numbers, not matrices. g(e1) should be a number. Which number?

 

[Tala.S]

1?

I'm still confused because we don't have g and I'm not sure what the method is to find g.

 

[Dick]

1?

I'm still confused because we don't have g and I'm not sure what the method is to find g.

Nope, you don't have g yet. We are still working on that. But you do know e1 is symmetric, so e1 is in U the kernel of g. What does that tell you about g(e1)?

 

[Tala.S]

But isn't e1 =[[1,0],[0,0]] ? How is this symmetric?

 

[Dick]

But isn't e1 =[[1,0],[0,0]] ? How is this symmetric?

I'm using a shorthand. I mean [[1,0],[0,0]] to be the matrix whose first row is 1,0 and the second row is 0,0. That's symmetric. Yes?

 

[Tala.S]

Is g(e1)=a

 

[Dick]

Is g(e1)=a

What is 'a'? What does 'kernel' mean? Explain 'kernel' in your own words.

 

[Tala.S]

Could the linear transformation be [0,1,-1,0] ?

 

[Dick]

Could the linear transformation be [0,1,-1,0] ?

It would if you can tell me what that means. I would like you to define g by telling me how g acts on a basis. What are g(e1), g(e2), g(e3) and g(e4)?

 

[Tala.S]

We need to find a linear transformation g: R^2x2 -> R that has U as kernel.
That means that all matrices of this type [a, b, b, c] will be transformed to zero since U is the kernel.
We need to find a linear transformation that 'sends' U to zero. Since we have two b's one can have a negative value and the other a positive value, a and c will be zero because we're going to multiply it with zero.
So no matter what the value of b is it will always be zero because +b-b = 0.

So our linear transformation can be :

[0,1,-1,0] or [0,-1,1,0]

?

 

[Dick]

We need to find a linear transformation g: R^2x2 -> R that has U as kernel.
That means that all matrices of this type [a, b, b, c] will be transformed to zero since U is the kernel.
We need to find a linear transformation that 'sends' U to zero. Since we have two b's one can have a negative value and the other a positive value, a and c will be zero because we're going to multiply it with zero.
So no matter what the value of b is it will always be zero because +b-b = 0.

So our linear transformation can be :

[0,1,-1,0] or [0,-1,1,0]

?

Absolutely right. But just writing [0,1,-1,0] doesn't say what you just told me. You mean that g([[a,b],[c,d]])=0*a+1*b+(-1)*c+0*d, right?

 

[Tala.S]

Yes that's what I mean :)

 

[Dick]

Yes that's what I mean :)

Great! Now try and think of ways to express yourself more clearly. Just saying '[0,1,-1,0]' is more of a puzzle than an answer.

 

[Tala.S]

Are you thinking about g(e1), g(e2), g(e3) and g(e4)?

 

[Dick]

Are you thinking about g(e1), g(e2), g(e3) and g(e4)?

I was thinking that in the general you don't explain enough when you write something down, but sure, speaking of those, writing [0,1,-1,0] doesn't tell me much. Wouldn't giving the values of g(e1), g(e2), g(e3) and g(e4) describe the tranformation more clearly?