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 with respect to the basis , we need to apply 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 basis function and calculate in terms of the given basis.
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Compute : So, .
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Compute : So, .
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Compute : So, .
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Compute : So, .
Matrix Representation
We can now form the matrix of with respect to the basis by using the coefficients of each 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.Ask a new question for Free
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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
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