Math Problem Statement
4.2b+Post-work.pdf
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Write the kernel as a span
Solution
To describe the kernel of the transformation as a span, we first need to understand the nature of the transformation. The given transformation maps a polynomial from the vector space (the space of polynomials of degree 2 or less) to the space of matrices.
The kernel of a linear transformation consists of all vectors in the domain that map to the zero vector in the codomain. In this case, the kernel will be the set of polynomials in that, when transformed by , result in the zero matrix.
The transformation is defined as:
p(0) & p(2) \\ p(2) & p(0) \end{bmatrix}$$ Where $$p(t) = a_2t^2 + a_1t + a_0$$. Let's proceed step by step: 1. **Find the kernel**: The kernel consists of the polynomials $$p(t)$$ such that $$T(p(t)) = \mathbf{0}$$, the zero matrix: $$T(p(t)) = \begin{bmatrix} p(0) & p(2) \\ p(2) & p(0) \end{bmatrix} = \begin{bmatrix} 0 & 0 \\ 0 & 0 \end{bmatrix}$$ This gives two conditions: $$p(0) = 0 \quad \text{and} \quad p(2) = 0.$$ For $$p(t) = a_2t^2 + a_1t + a_0$$, these conditions are: $$p(0) = a_0 = 0,$$ $$p(2) = a_2(2)^2 + a_1(2) + a_0 = 4a_2 + 2a_1 + a_0 = 0.$$ Substituting $$a_0 = 0$$ into the second condition: $$4a_2 + 2a_1 = 0.$$ Simplifying: $$2a_2 + a_1 = 0 \quad \text{or} \quad a_1 = -2a_2.$$ 2. **Kernel description**: Thus, the polynomials in the kernel are of the form: $$p(t) = a_2t^2 + a_1t + a_0 = a_2t^2 - 2a_2t.$$ Factoring out $$a_2$$: $$p(t) = a_2(t^2 - 2t).$$ Therefore, the kernel is the span of the polynomial $$t^2 - 2t$$: $$\text{ker}(T) = \text{span}\{ t^2 - 2t \}.$$ So, the kernel of $$T$$ is spanned by $$\{ t^2 - 2t \}$$. Would you like a more detailed explanation of the other parts of the problem, or is there anything you'd like to explore further? Here are a few related questions for your reference: 1. How would you compute the range of a linear transformation? 2. What is the general form of a linear transformation from one vector space to another? 3. Can the kernel of a transformation ever be the entire domain? 4. What properties must a transformation have to be considered linear? 5. How do we check if a set of polynomials forms a basis for a vector space? **Tip:** Always check the conditions for the kernel by substituting values into the transformation equation to solve for the required vector space.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Transformation
Kernel of a Transformation
Polynomials
Matrix Representation
Formulas
T(p(t)) = [p(0) p(2); p(2) p(0)]
Theorems
Kernel of a Linear Transformation
Suitable Grade Level
College Level (Math 231 or equivalent)
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