Define ƒ : P₂ → R² by ƒ(ax² + bx + c) = (a + c) · (a) Find the nullspace of f. (b) Find the nullity of f. (c) Find the range space of f. (Write it as a span) (d) What is the rank of f?
Define ƒ : P₂ → R² by ƒ(ax² + bx + c) = (a + c) · (a) Find the nullspace of f. (b) Find the nullity of f. (c) Find the range space of f. (Write it as a span) (d) What is the rank of f?
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Transcribed Image Text:1. Define \( f : \mathcal{P}_2 \to \mathbb{R}^2 \) by \( f(ax^2 + bx + c) = \begin{pmatrix} a+b \\ a-c \end{pmatrix} \).
(a) Find the nullspace of \( f \).
(b) Find the nullity of \( f \).
(c) Find the range space of \( f \). (Write it as a span)
(d) What is the rank of \( f \)?
![**Exercise: Linear Transformations and Matrix Representation**
(e) Use a nullity argument to determine whether \( f \) is injective.
(f) Use a rank argument to determine whether \( f \) is surjective.
(g) Find a matrix representation of \( f : \mathcal{P}_2 \rightarrow \mathbb{R}^2 \) by
\[
f(ax^2 + bx + c) = \begin{pmatrix} a + b \\ a - c \end{pmatrix}
\]
(h) Find a basis for the Column Space of this matrix.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc495729c-fc87-4b7b-a937-12ad0926c14e%2F6e1f7d5b-556b-47b4-8de7-387324f11a96%2Fsikt6sp_processed.png&w=3840&q=75)
Transcribed Image Text:**Exercise: Linear Transformations and Matrix Representation**
(e) Use a nullity argument to determine whether \( f \) is injective.
(f) Use a rank argument to determine whether \( f \) is surjective.
(g) Find a matrix representation of \( f : \mathcal{P}_2 \rightarrow \mathbb{R}^2 \) by
\[
f(ax^2 + bx + c) = \begin{pmatrix} a + b \\ a - c \end{pmatrix}
\]
(h) Find a basis for the Column Space of this matrix.
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