text: # Matrix representations of linear transformations
  
  • narfarnst
  In summary, the given problem involves two transformations, T and U, and their composition UT. The standard bases for P2 and R3 are used to compute the matrix representation of U ([U]yB) and T ([T]yB). The transformed bases for U and T are obtained by applying the given transformations to the standard bases. The matrix representation of U is [1 1 0; 0 0 1; 1 -1 0] and the matrix representation of T is [2 3 0; 0 3 4; 0 0 0]. To obtain the composition UT, the matrix representation of U is multiplied by the matrix representation of T.
  narfarnst
  
  ## Homework Statement
  
  Let g(x)=3+x and T(f(x))=f'(x)g(x)+2f(x), and U(a+bx+cx2)=(a+b,c,a-b). So T:P2(R)-->P2(R) and U:P2(R)-->R3.
  And let B and y be the standard ordered bases for P2 and R3 respectively.
  Compute the matrix representation of U (denoted yB) and T ([T]yB) and their composition UT.
  
  None, really.
  
  ## The Attempt at a Solution
  
  So I get what to do here, I'm just a little hung up with the polynomials.
  First, you use the transformations given and the standard bases, and transform the standard bases. B={1,x,x2} and y={(1,0,0), (0,1,0), (0,0,1)}.
  So U(1)=(1,0,1), U(x)=(1,0,-1), and U(x2)=(0,1,0).
  And T(1)=2, T(x)=3+3x, and T(x2)=6x+4x2.
  
  Now, I'm confused as to how I right the actual matrix of transformation for these. I know what when you're using just numbers (T:R3-->R2 for example), you write your transformed B basis in terms of coefficients of your y basis. But I'm not sure how to do that here.
  
  Do I write T(1),T(x), and T(x2) in terms of coefficients of y? Or am I supposed to do this for U?Any help on how to go about this would be really appreciated. Thanks!
  
  ## 1. What is a matrix representation of a linear transformation?
  
  A matrix representation of a linear transformation is a way to represent a linear transformation using a matrix. It is a way to visualize and perform calculations on a linear transformation using the properties and operations of matrices.
  
  ## 2. How do you create a matrix representation of a linear transformation?
  
  To create a matrix representation of a linear transformation, you need to choose a basis for both the domain and the codomain of the transformation. Then, you apply the transformation to each basis vector and record the resulting coordinates in the matrix. The columns of the matrix will represent the coordinates of the transformed basis vectors in the codomain.
  
  ## 3. What is the significance of the standard basis when creating a matrix representation?
  
  The standard basis is a set of basis vectors that have a value of 1 in one coordinate and 0 in all other coordinates. When creating a matrix representation, the standard basis is commonly used because it simplifies the process and allows for easy visualization and manipulation of the matrix.
  
  ## 4. Can a linear transformation have more than one matrix representation?
  
  Yes, a linear transformation can have multiple matrix representations, depending on the choice of bases for the domain and codomain. However, all matrix representations of the same linear transformation will have the same rank and determinant.
  
  ## 5. How do you use matrix representations to perform calculations on linear transformations?
  
  Matrix representations can be used to perform calculations on linear transformations by using the properties of matrices. For example, matrix multiplication can be used to combine multiple linear transformations, and the determinant of a matrix representation can be used to determine if the transformation is invertible.
  
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