Consider a constant vector B e R*. Is the (projection) function F(ā)=| a-B à - dot products – where äeR“ a linear transformation? Justify, always.
Consider a constant vector B e R*. Is the (projection) function F(ā)=| a-B à - dot products – where äeR“ a linear transformation? Justify, always.
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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![**Problem Statement:**
Consider a constant vector \(\vec{b} \in \mathbb{R}^4\). Is the (projection) function
\[
F(\vec{a}) = \left(\frac{\vec{a} \cdot \vec{b}}{\vec{a} \cdot \vec{a}}\right) \vec{a}
\]
where \(\vec{a} \in \mathbb{R}^4\) a linear transformation? Justify, always.
**Explanation:**
In this problem, you are tasked with determining whether the function \(F(\vec{a})\), which projects vector \(\vec{b}\) onto vector \(\vec{a}\), is a linear transformation.
To do so, we must verify whether \(F\) satisfies the properties of a linear transformation:
1. **Additivity:** \(F(\vec{a}_1 + \vec{a}_2) = F(\vec{a}_1) + F(\vec{a}_2)\)
2. **Homogeneity:** \(F(c \cdot \vec{a}) = c \cdot F(\vec{a})\) for any scalar \(c\).
Analyzing these properties using the given formula will help you determine the linearity of this function.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F80a13d8c-5188-4010-ba23-710cc489cc25%2Fe35d2ca4-79b6-4e65-8e5e-f17c4e0d6fae%2F60qvrvc_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Consider a constant vector \(\vec{b} \in \mathbb{R}^4\). Is the (projection) function
\[
F(\vec{a}) = \left(\frac{\vec{a} \cdot \vec{b}}{\vec{a} \cdot \vec{a}}\right) \vec{a}
\]
where \(\vec{a} \in \mathbb{R}^4\) a linear transformation? Justify, always.
**Explanation:**
In this problem, you are tasked with determining whether the function \(F(\vec{a})\), which projects vector \(\vec{b}\) onto vector \(\vec{a}\), is a linear transformation.
To do so, we must verify whether \(F\) satisfies the properties of a linear transformation:
1. **Additivity:** \(F(\vec{a}_1 + \vec{a}_2) = F(\vec{a}_1) + F(\vec{a}_2)\)
2. **Homogeneity:** \(F(c \cdot \vec{a}) = c \cdot F(\vec{a})\) for any scalar \(c\).
Analyzing these properties using the given formula will help you determine the linearity of this function.
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