Help with Proof of Junghenn Proposition 9.2.3 - A Course in Real Analysis

[Math Amateur]

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:
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In the above proof Junghenn let's \(\displaystyle \mathbf{a}_i = ( a_{i1}, a_{i2}, \ ... \ ... \ , a_{in} ) \)

and then states that \(\displaystyle T \mathbf{x} = ( \mathbf{a}_1 \cdot \mathbf{x}, \mathbf{a}_2 \cdot \mathbf{x}, \ ... \ ... \ , \mathbf{a}_n \cdot \mathbf{x} )\) where \(\displaystyle \mathbf{x} = ( x_1, x_2, \ ... \ ... \ x_n )\)(Note: Junghenn defines vectors in \mathbb{R}^n as row vectors ... ... )Now I believe I can show \(\displaystyle T \mathbf{x}^t = [a_{ij} ]_{ m \times n } \mathbf{x}^t = ( \mathbf{a}_1 \cdot \mathbf{x}, \mathbf{a}_2 \cdot \mathbf{x}, \ ... \ ... \ , \mathbf{a}_n \cdot \mathbf{x} )^t\) ...... ... as follows:
\(\displaystyle T \mathbf{x}^t = [a_{ij} ]_{ m \times n } \mathbf{x}^t = \begin{pmatrix} a_{11} & a_{12} & ... & ... & a_{1n} \\ a_{21} & a_{22} & ... & ... & a_{2n} \\ ... & ... & ... & ... & ... \\ ... & ... & ... & ... & ... \\ a_{m1} & a_{m2} & ... & ... & a_{mn} \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ . \\ . \\ x_n \end{pmatrix}\)
\(\displaystyle = \begin{pmatrix} a_{11} x_1 + a_{12} x_2 + \ ... \ ... \ + a_{1n} x_n \\ a_{21} x_1 + a_{22} x_2 + \ ... \ ... \ + a_{2n} x_n \\ ... \\ ... \\ a_{m1} x_1 + a_{m2} x_2 + \ ... \ ... \ + a_{mn} x_n \end{pmatrix} \)
\(\displaystyle = \begin{pmatrix} \mathbf{a}_1 \cdot \mathbf{x} \\ \mathbf{a}_2 \cdot \mathbf{x} \\ . \\ . \\ \mathbf{a}_n \cdot \mathbf{x} \end{pmatrix}\)
\(\displaystyle = ( \mathbf{a}_1 \cdot \mathbf{x}, \mathbf{a}_2 \cdot \mathbf{x}, \ ... \ ... \ , \mathbf{a}_n \cdot \mathbf{x} )^t \)

So ... I have shown\(\displaystyle T \mathbf{x}^t = [a_{ij} ]_{ m \times n } \mathbf{x}^t = ( \mathbf{a}_1 \cdot \mathbf{x}, \mathbf{a}_2 \cdot \mathbf{x}, \ ... \ ... \ , \mathbf{a}_n \cdot \mathbf{x} )^t\) ...How do I reconcile or 'square' that with Junghenn's statement that \(\displaystyle T \mathbf{x} = ( \mathbf{a}_1 \cdot \mathbf{x}, \mathbf{a}_2 \cdot \mathbf{x}, \ ... \ ... \ , \mathbf{a}_n \cdot \mathbf{x} )\) where \(\displaystyle \mathbf{x} = ( x_1, x_2, \ ... \ ... \ x_n )\)(Note: I don't think that taking the transpose of both sides works ... ?)
Hope someone can help ...

Peter

 

[Opalg]

Junghenn defines the relation between the linear transformation $T$ and the matrix $A$ by \(\displaystyle T \mathbf{x} = ( \mathbf{a}_1 \cdot \mathbf{x},\, \mathbf{a}_2 \cdot \mathbf{x}, \ldots , \mathbf{a}_n \cdot \mathbf{x} )\) where \(\displaystyle \mathbf{x} = ( x_1,\, x_2, \ldots, x_n )\). This – as you show – is equivalent to the statement $(T\mathbf{x})^t = A\mathbf{x}^t.$

In other words, linear transformations act on elements of $\mathbb{R}^n$ (which Junghenn defines as row vectors), but matrices act (by pre-multiplication) on column vectors. There is no great mathematical significance in this. Junghenn probably prefers row vectors simply for convenience, because they take up less room on the printed page. But the $m\times n$ matrix $A$ has to be multiplied by an $n\times1$ vector (in other words, a column vector) in order for the matrix multiplication to be defined.

So if you are talking about linear transformations, you need to use row vectors, but if you want to deal with their associated matrices then you must use column vectors.

 

[Math Amateur]

Junghenn defines the relation between the linear transformation $T$ and the matrix $A$ by \(\displaystyle T \mathbf{x} = ( \mathbf{a}_1 \cdot \mathbf{x},\, \mathbf{a}_2 \cdot \mathbf{x}, \ldots , \mathbf{a}_n \cdot \mathbf{x} )\) where \(\displaystyle \mathbf{x} = ( x_1,\, x_2, \ldots, x_n )\). This – as you show – is equivalent to the statement $(T\mathbf{x})^t = A\mathbf{x}^t.$

In other words, linear transformations act on elements of $\mathbb{R}^n$ (which Junghenn defines as row vectors), but matrices act (by pre-multiplication) on column vectors. There is no great mathematical significance in this. Junghenn probably prefers row vectors simply for convenience, because they take up less room on the printed page. But the $m\times n$ matrix $A$ has to be multiplied by an $n\times1$ vector (in other words, a column vector) in order for the matrix multiplication to be defined.

So if you are talking about linear transformations, you need to use row vectors, but if you want to deal with their associated matrices then you must use column vectors.

Thanks Opalg ...

To know that the representation of vectors varies according to context like that is important to me in fully understanding what is going on in the various proofs/results in Euclidean and metric spaces ...

Thanks again for that post!

Peter