Linear Transformations (polynomials/matrices)

In summary, the problem involves finding the formula for TS(p(x)) where S is a linear transformation from P2 into P3 over R and T is a linear transformation from P3 over R into R2x2. The transition matrix for S is found using the standard basis of P2 and the identity matrix is used for the transition matrix of T. Multiplying the two matrices yields [0,0,0;1,0,0;0,1,0;0,0,1]. To find the formula for TS(p(x)), it is necessary to understand that the vector [0,a0;a1,a2] represents the terms of p(x) and the mapping of P2 into R2x2. Further
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jesuslovesu
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[SOLVED] Linear Transformations (polynomials/matrices)

Never mind, I can see it now, thanks

Homework Statement


Let S be the linear transformation on P2 into P3 over R. S(p(x)) = xp(x)
Let T be the linear transformation on P3 over R into R2x2 defined by T(a0 + a1x + a2x^2 + a3x^3) = [ a0 a1; a2 a3]

Find a formula for TS(p(x)).

The Attempt at a Solution


The first thing I do is find the S(A) where A is the standard basis of P2 and I place that into a transition matrix from the basis B (std. basis of P3).
B,A = [0,0,0;1,0,0;0,1,0;0,0,1]
Then I do the similar steps for fining [T]C,B where C = E2x2
[T]C,B = I4 (identity matrix of a 4x4)
Multiplying the matrix yields: [T]* = [0,0,0;1,0,0;0,1,0;0,0,1]

I am fairly positive that the math up to this point is accurate. (I get the correct range of T).

My question is how do I specifically find the formula for TS(p(x)) using the last matrix that I found? I know it's a mapping of P2 into R2x2, but I don't quite see how they get [0, a0; a1, a2] as the matrix. I know it lines up with TS, but still
 
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don't understand how they get the vector to be made up of only the terms of p(x). Any help would be greatly appreciated!
 

What are linear transformations?

Linear transformations refer to the process of transforming a mathematical object, such as a polynomial or a matrix, by multiplying it with a constant value and adding it to another mathematical object. This process results in a new object that is a linear combination of the original objects.

How do linear transformations work?

Linear transformations work by applying a set of mathematical operations, such as multiplication and addition, to a given object. These operations are typically represented by a matrix or a polynomial, and they result in a new object that is a linear combination of the original object.

What is the purpose of using linear transformations?

The purpose of using linear transformations is to transform a given mathematical object into a new one that is easier to work with or has more desirable properties. This can be useful in various mathematical applications, such as solving systems of equations, optimizing functions, and analyzing data.

What are some common types of linear transformations?

Some common types of linear transformations include scaling, translation, rotation, and reflection. Scaling involves multiplying an object by a constant value, translation involves adding a constant value, rotation involves rotating an object around a fixed point, and reflection involves reflecting an object across a line or plane.

How are linear transformations related to matrices and polynomials?

Linear transformations can be represented by matrices and polynomials. The coefficients of the matrix or polynomial determine the specific transformation being applied. Additionally, matrices and polynomials can be used to perform linear transformations on other mathematical objects.

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