Norm of a Linear Transformation: Proving Homogeneity From Definition - Peter

In summary, the conversation revolves around Chapter 9 of Hugo D. Junghenn's book "A Course in Real Analysis" and the proof of Proposition 9.2.3. The question is raised about how to demonstrate the homogeneity property, specifically showing that $\| tT \| = |t| \| T \|$. The answer is provided by explaining that the defining properties for the norm in $\mathcal{L}(\Bbb{R}^n,\Bbb{R}^m)$ follow from those in $\Bbb{R}^m$, and by taking the supremum over all $\mathbf{x}$ with $\|\mathbf{x}\|=1$, it can be shown that $\|t
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I am reading Hugo D. Junghenn's book: "A Course in Real Analysis" ...

I am currently focused on Chapter 9: "Differentiation on \(\displaystyle \mathbb{R}^n\)"

I need some help with the proof of Proposition 9.2.3 ...

Proposition 9.2.3 and the preceding relevant Definition 9.2.2 read as follows:
View attachment 7904
https://www.physicsforums.com/attachments/7905

Near the end of the above proof we read the following:

" ... ... To see that \(\displaystyle \| T \|\) defines a norm, note that homogeneity follows directly from the definition ... ... "
My question is as follows:

How, exactly, do we demonstrate rigorously that homogeneity follows directly from the definition ... that is how do we show that

\(\displaystyle \| t T \| = \ \mid t \mid \| T \|\) ... ... for \(\displaystyle t \in \mathbb{R}\) ... ... ?

Help will be appreciated ...

Peter
 
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  • #2
Peter said:
How, exactly, do we demonstrate rigorously that homogeneity follows directly from the definition ... that is how do we show that

\(\displaystyle \| t T \| = \ \mid t \mid \| T \|\) ... ... for \(\displaystyle t \in \mathbb{R}\) ... ... ?
The defining properties for the norm in $\mathcal{L}(\Bbb{R}^n,\Bbb{R}^m)$ follow from those for the norm in $\Bbb{R}^m$.

For the homogeneity property, the definition of $tT$ is given by $(tT)\mathbf{x}) = t(T\mathbf{x})$. So $$\|(tT)\mathbf{x})\| = \|t(T\mathbf{x})\| = |t|\|T\mathbf{x}\|.$$ If you now take the sup over all $\mathbf{x}$ with $\|\mathbf{x}\|=1$, it follows that $\|tT\| = |t|\|T\|.$
 
  • #3
Opalg said:
The defining properties for the norm in $\mathcal{L}(\Bbb{R}^n,\Bbb{R}^m)$ follow from those for the norm in $\Bbb{R}^m$.

For the homogeneity property, the definition of $tT$ is given by $(tT)\mathbf{x}) = t(T\mathbf{x})$. So $$\|(tT)\mathbf{x})\| = \|t(T\mathbf{x})\| = |t|\|T\mathbf{x}\|.$$ If you now take the sup over all $\mathbf{x}$ with $\|\mathbf{x}\|=1$, it follows that $\|tT\| = |t|\|T\|.$
Thanks Opalg ...

That cleared up that issue ...

Thanks again ...

Peter
 

Related to Norm of a Linear Transformation: Proving Homogeneity From Definition - Peter

1. What is the norm of a linear transformation?

The norm of a linear transformation is a measure of the size or magnitude of the transformation. It is a real-valued function that maps a vector space to the non-negative real numbers. It is used to quantify the amount by which the transformation stretches or shrinks a vector.

2. How is the norm of a linear transformation defined?

The norm of a linear transformation is defined as the maximum amount by which the transformation stretches any unit vector. In other words, it is the largest value that the transformation can take on when applied to any unit vector in the vector space.

3. What is homogeneity in the context of linear transformations?

Homogeneity in the context of linear transformations means that the transformation preserves the scaling of vectors. In other words, if a vector is multiplied by a scalar, the resulting vector after transformation is also multiplied by the same scalar. This property is also known as the scaling property.

4. How can we prove homogeneity of a linear transformation?

To prove homogeneity of a linear transformation, we need to show that the transformation satisfies the scaling property for all possible vectors and scalars. This can be done by using the definition of a linear transformation and showing that it holds for any vector and scalar in the vector space. Alternatively, we can use the properties of a norm to show that the norm of the transformed vector is equal to the norm of the original vector multiplied by the scalar.

5. Why is proving homogeneity important in linear transformations?

Proving homogeneity is important in linear transformations because it is one of the fundamental properties that define a linear transformation. Without homogeneity, the transformation would not preserve the scaling of vectors, which is a crucial aspect of linear transformations. Additionally, proving homogeneity allows us to use various properties of a norm to simplify calculations and make proofs more manageable.

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