Math Problem Statement

Let V = {f \in K[x] : deg(f) \leq 3} and let T : V \rightarrow V with Tf(x) = f(x+1), compute the Matrix of T with respect to the basis (f_0, f_1, f_2, f_3) = (1,x,x^2,x^3).

Solution

To compute the matrix of the linear transformation TT with respect to the basis (f0,f1,f2,f3)=(1,x,x2,x3)(f_0, f_1, f_2, f_3) = (1, x, x^2, x^3), we need to apply TT to each basis element and express the result as a linear combination of the basis elements.

Step-by-Step Solution

Let’s go through each fif_i basis function and calculate T(fi)=fi(x+1)T(f_i) = f_i(x+1) in terms of the given basis.

  1. Compute T(f0)T(f_0): f0(x)=1T(f0)=f0(x+1)=1f_0(x) = 1 \Rightarrow T(f_0) = f_0(x+1) = 1 So, T(f0)=1f0+0f1+0f2+0f3T(f_0) = 1 \cdot f_0 + 0 \cdot f_1 + 0 \cdot f_2 + 0 \cdot f_3.

  2. Compute T(f1)T(f_1): f1(x)=xT(f1)=f1(x+1)=x+1f_1(x) = x \Rightarrow T(f_1) = f_1(x+1) = x + 1 So, T(f1)=1f0+1f1+0f2+0f3T(f_1) = 1 \cdot f_0 + 1 \cdot f_1 + 0 \cdot f_2 + 0 \cdot f_3.

  3. Compute T(f2)T(f_2): f2(x)=x2T(f2)=f2(x+1)=(x+1)2=x2+2x+1f_2(x) = x^2 \Rightarrow T(f_2) = f_2(x+1) = (x+1)^2 = x^2 + 2x + 1 So, T(f2)=1f0+2f1+1f2+0f3T(f_2) = 1 \cdot f_0 + 2 \cdot f_1 + 1 \cdot f_2 + 0 \cdot f_3.

  4. Compute T(f3)T(f_3): f3(x)=x3T(f3)=f3(x+1)=(x+1)3=x3+3x2+3x+1f_3(x) = x^3 \Rightarrow T(f_3) = f_3(x+1) = (x+1)^3 = x^3 + 3x^2 + 3x + 1 So, T(f3)=1f0+3f1+3f2+1f3T(f_3) = 1 \cdot f_0 + 3 \cdot f_1 + 3 \cdot f_2 + 1 \cdot f_3.

Matrix Representation

We can now form the matrix of TT with respect to the basis (f0,f1,f2,f3)(f_0, f_1, f_2, f_3) by using the coefficients of each T(fi)T(f_i) in terms of the basis elements:

\begin{pmatrix} 1 & 0 & 0 & 0 \\ 1 & 1 & 0 & 0 \\ 1 & 2 & 1 & 0 \\ 1 & 3 & 3 & 1 \\ \end{pmatrix}$$ This matrix represents the transformation $$T$$ in the given basis. Would you like further details on any of these steps, or do you have any questions? --- ### Related Questions 1. What is the interpretation of the matrix $$T$$ in terms of shifting polynomial functions? 2. How does changing the basis affect the matrix representation of $$T$$? 3. Can this method be generalized for transformations involving other shifts, such as $$f(x+2)$$? 4. What properties (e.g., eigenvalues, eigenvectors) does the matrix $$T$$ have? 5. How would the matrix of $$T$$ change if we increased the degree bound on $$V$$? #### Tip When working with transformations involving polynomial shifts, expressing the result in terms of binomial expansions can often simplify the calculation process.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Linear Transformation
Matrix Representation
Polynomial Basis

Formulas

Matrix representation of T(f) in terms of basis (f_0, f_1, f_2, f_3)
Expansion of (x+1)^n for polynomial functions

Theorems

Basis representation theorem
Binomial theorem for polynomial expansions

Suitable Grade Level

Undergraduate Mathematics