Standard Ordered Basis : Change of Basis

Please see the attached file for full problem description. Let E= {1, x, x^2, x^3} be the standard ordered basis for the space P_3. Show that G= {1+x, 1-x, 1-x^2, 1-x^3} is also a basis for P_3, and write the change of basis matrix S from G to E. Show work.

Change of basis

Please see the attached file for full problem description.

Change of Basis : Matrix Representation

Please see the attached file for full problem description. Let E= {1, x, x^2, x^3} be the standard ordered basis for the space P_3. G= {1+x, 1-x, 1-x^2, 1-x^3} is also a basis for P_3. Write the matrix representation [p(x)]_G for the polynomial (vector) p(x)=3x^3-2x+4 from P_3 with respect to the basis G. Show work.

Matrix Representation: Linear Transformation

Please see the attached file for full problem description. Let T be a linear operator on P_3 defined as follows: T(ax^3 + bx^2 + cx + d) = (a – b)x^2 + (c – d)x + (a + b – c). Write the matrix [T]_G which represents T with respect to the basis G which = {1 + x, 1 – x, 1 – x^2, 1 – x^3}. Show work.

Matrix Representation : Linear Transformation

Please see the attached file for full problem description. Let T be a linear operator on P_3 defined as follows: T(ax^3 + bx^2 + cx + d) = (a – b)x^2 + (c – d)x + (a + b – c). The matrix [T]_G which represents T with respect to the basis G which = {1 + x, 1 – x, 1 – x^2, 1 – x^3}. Show that the standard matrix representation and the preceding matrix representation are similar. Show work.