Matrices of linear transformations

In summary, the linear operator T: P2 - P2 is defined by T(a0 + a1x + a2x2) = a0 + a1(x - 1) + a2(x - 1)2. To find the matrix for T with respect to the standard basis B = {1, x, x2}, we can use the equation [T]B[x]B = [T(x)]B and solve for the components of x and x2 in the basis {1,x,x2}. The components of x are (0,1,0) and the components of x2 are (0,0,1).
  • #1
derryck1234
56
0

Homework Statement



Let T: P2 - P2 be the linear operator defined by
T(a0 + a1x + a2x2) = a0 + a1(x - 1) + a2(x - 1)2

(a) Find the matrix for T with respect to the standard basis B = {1, x, x2}.

Homework Equations



[T]B[x]B = [T(x)]B

The Attempt at a Solution



T(1) = a0 + a1(1 - 1) + a2(1 - 1)2
= a0

Okay...I don't know if I'm going in the right direction here?...and if I am, I'm not sure how to obtain vectors from the proceeding expressions for T(x) and T(x2)??

Any help would be appreciated.

Thanks.

Derryck.
 
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  • #2
Hi Derryck! :smile:
derryck1234 said:
Let T: P2 - P2 be the linear operator defined by
T(a0 + a1x + a2x2) = a0 + a1(x - 1) + a2(x - 1)2

(a) Find the matrix for T with respect to the standard basis B = {1, x, x2}.
…
T(1) = a0 + a1(1 - 1) + a2(1 - 1)2
= a0

nooo :redface: …

suppose it was T(a0 + a1x + a2x2) = a0 + a1(x + 1) + a2(x + 1)2

then your method would give you T(1) = a0 + a1(1 + 1) + a2(1 + 1)2
= a0 + 2a1 + 4a2

but the correct way is to start by saying 1 = 1 + 0x + 0x2, so T(1) = … :wink:

carry on from there :smile:​
 
  • #3
Wo...?

But then is this the same as saying (in vector form):

1 1
0 = 0
0 0 ?

I'm not sure what you mean by 1 = 1 + 0x + 0x2, because what I see from this, I would think that it would be more correct to say:

1 = a0 + a1x + a2x2, then to solve...? Flip, I'm so bad at this I know...but please, just humor me a little...it will help a great deal...

Cheers

Derryck
 
  • #4
derryck1234 said:
I'm not sure what you mean by 1 = 1 + 0x + 0x2, because what I see from this, I would think that it would be more correct to say:

1 = a0 + a1x + a2x2, then to solve...?

Yes, and the solution is a0 = 1, a1 = a2 = 0, isn't it? :wink:

In other words: in the basis {1,x,x2}, 1 has components (1,0,0).

ok, now what are the components of x ?

and what are the components of x2 ? :smile:​
 
  • #5
Ok, the components of x would be found by solving:

x = a0 + a1(x - 1) + a2(x - 1)2

So I would then say

a1(x - 1) = x
Therefore a1 = x / (x - 1)... Am I going along the right lines? I'm not too sure what to do from here.

Thanks for the help so far though really!

Cheers

Derryck
 
  • #6
derryck1234 said:
Ok, the components of x would be found by solving:

x = a0 + a1(x - 1) + a2(x - 1)2

No!

In the basis {1,x,x2}, what is x?

x = ?1 + ?x + ?x2
 

FAQ: Matrices of linear transformations

What is a matrix of linear transformations?

A matrix of linear transformations is a way to represent a linear transformation, which is a function that maps points in one vector space to points in another vector space while preserving the structure of the vector space. It is a rectangular array of numbers that can be used to represent the transformation in a more concise and efficient way.

How is a matrix of linear transformations used in mathematics?

Matrices of linear transformations are used extensively in mathematics, especially in the fields of linear algebra and multivariable calculus. They are used to solve systems of linear equations, find eigenvalues and eigenvectors, and represent geometric transformations such as rotations, reflections, and scaling.

What are the properties of a matrix of linear transformations?

A matrix of linear transformations has several important properties, including linearity, associativity, and distributivity. Linearity means that the transformation preserves addition and scalar multiplication. Associativity means that the order in which the transformations are applied does not change the result. Distributivity means that the transformation can be distributed over addition and subtraction.

How is a matrix of linear transformations different from a regular matrix?

A matrix of linear transformations is different from a regular matrix in that it represents a linear transformation rather than a set of numbers. While a regular matrix can be used for various operations, a matrix of linear transformations is specifically used for representing and performing linear transformations.

Can a matrix of linear transformations have a determinant?

Yes, a matrix of linear transformations can have a determinant. The determinant of a matrix of linear transformations is a real number that represents the scaling factor of the transformation. It is equal to the product of the eigenvalues of the matrix and can be used to determine properties such as invertibility and orientation of the transformation.

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