Linear Maps in $\mathbb{R}^n$ and $\mathbb{R}^m$: Proving Properties with $M$

[mathmari]

Hey!

Let $1\leq n,m\in \mathbb{N}$ and let $\phi, \psi:\mathbb{R}^n\rightarrow \mathbb{R}^m$ be linear maps. Let $\lambda\in \mathbb{R}$.

Show the following:

1. $M(\phi +\psi )=M(\phi )+M(\psi )$
2. $M(\lambda \phi )=\lambda M(\phi )$

What exactly is $M$, it is not defined in this exercise? Is it a matrix? (Wondering)

 

[I like Serena]

Hey mathmari!

I don't know what M is either. It doesn't look like a matrix. (Worried)

I think it must have been defined somewhere in your text.

 

[mathmari]

I don't know what M is either. It doesn't look like a matrix. (Worried)

I think it must have been defined somewhere in your text.

Can we show the properties if we suppose that $M$ is a map? (Wondering)

 

[I like Serena]

Can we show the properties if we suppose that $M$ is a map? (Wondering)

$M$ must be a function that maps functions to some vector space so that addition and multiplication with a real scalar are defined.

Suppose we assume $M$ is an arbitrary map.
Can we find a counter example so that $M$ is not linear? (Wondering)

Suppose we pick $\phi: x\mapsto 0$ and $\psi: x\mapsto 0$.
And suppose we pick $M: (\mathbb R^n\to\mathbb R^m)\to \mathbb R$ given by $\chi \mapsto 1$.
Are the linearity conditions satisfied? (Wondering)

 

[mathmari]

$M$ must be a function that maps functions to some vector space so that addition and multiplication with a real scalar are defined.

Suppose we assume $M$ is an arbitrary map.
Can we find a counter example so that $M$ is not linear? (Wondering)

Suppose we pick $\phi: x\mapsto 0$ and $\psi: x\mapsto 0$.
And suppose we pick $M: (\mathbb R^n\to\mathbb R^m)\to \mathbb R$ given by $\chi \mapsto 1$.
Are the linearity conditions satisfied? (Wondering)

I got stuck right now, how is the map defined? (Wondering) I saw now that the title of this exercise is "Linear Maps and Matrices". So is maybe $M$ related to matrices? (Wondering)

 

[I like Serena]

I got stuck right now, how is the map defined?

Let's define $L$ as the set of linear functions in $\mathbb R^n\to\mathbb R^m$, so that $\phi, \psi\in L$.
An example of an element $\phi\in L$ is the function given by $\phi(\mathbf x)=\mathbf 0$ that maps any $\mathbf x\in\mathbb R^n$ to the zero element $\mathbf 0\in \mathbb R^m$ yes? (Thinking)

Now $M$ must be a function that has an element like $\phi\in L$ as input.
And its output must be something that we can add, and that we can multiply with a real scalar.
Let's define the co-domain of $M$ as $V$ for which addition and scalar multiplication are defined.
Then $M: L\to V$, isn't it? (Wondering)

As an example, let's pick again $\phi\in L$ given by $\phi(\mathbf x)=\mathbf 0$.
Then $M(\phi)$ must be an element of $V$ yes? (Thinking)
Let $\mathbf v$ be a non-zero element in $V$.
Continuing the example, we can pick $M(\phi)=\mathbf v$, can't we? (Wondering)
Moreover, we can choose $M$ such that for any $\chi\in L$ we have $M(\chi)=\mathbf v\ne\mathbf 0$.

If we pick this $M$, and suppose we have some $\phi,\psi\in L$, what will $M(\phi)$, $M(\psi)$, and $M(\phi+\psi)$ be? (Wondering)

I saw now that the title of this exercise is "Linear Maps and Matrices". So is maybe $M$ related to matrices?

Since $M$ is applied to a function and not to a vector, it seems to me that it cannot be a matrix.
It is a 'Map' though, and the question asks to prove that it is a Linear Map. (Thinking)

 

[mathmari]

Since $M$ is applied to a function and not to a vector, it seems to me that it cannot be a matrix.
It is a 'Map' though, and the question asks to prove that it is a Linear Map. (Thinking)

I think $M$ is a transformation matrix with respect to the canonical basis. What do we have in this case then? (Wondering)

$M$ is then a linear map, or not? (Wondering)

Then we get the following:

1. For $x\in \mathbb{R}^n$ we have the following: $$(M(\phi +\psi ))(x)=M(\phi +\psi )(x) \ \overset{\phi, \psi \text{ linear }}{ = } \ M(\phi(x) +\psi(x) ) \ \overset{M \text{ linear }}{ = } \ M(\phi(x)) +M(\psi(x) )=(M(\phi ))(x)+(M(\psi ))(x)$$
   
2. For $x\in \mathbb{R}^n$ we have the following: $$(M(\lambda \phi ))(x)=M(\lambda \phi )(x) \ \overset{\phi\text{ linear }}{ = } \ M(\lambda \phi(x) ) \ \overset{M \text{ linear }}{ = } \ \lambda M( \phi(x) )=\lambda (M(\phi ))(x)$$

Is everything correct? (Wondering)

 

[I like Serena]

I think $M$ is a transformation matrix with respect to the canonical basis. What do we have in this case then? (Wondering)

$M$ is then a linear map, or not? (Wondering)

Then we get the following:

1. For $x\in \mathbb{R}^n$ we have the following: $$(M(\phi +\psi ))(x)=M(\phi +\psi )(x) \ \overset{\phi, \psi \text{ linear }}{ = } \ M(\phi(x) +\psi(x) ) \ \overset{M \text{ linear }}{ = } \ M(\phi(x)) +M(\psi(x) )=(M(\phi ))(x)+(M(\psi ))(x)$$

If we treat $M$ as a matrix that represents a linear transformation, and if we take $M(\phi)$ to mean the composition $M\circ\phi$, we get indeed:
$$(M(\phi +\psi ))(x)=(M\circ(\phi +\psi ))(x) = M((\phi +\psi )(x))$$

It does not follow from the linearity condition that:
$$M((\phi +\psi )(x)) \ \overset{\phi, \psi \text{ linear }}{ = } \ M(\phi(x) +\psi(x) )$$
though. (Worried)

Instead it follows from how function addition is usually defined.
That is, functions are added pointwise.
The function $(\phi+\psi)$ is such that for all $x$ we have that $(\phi+\psi)(x)=\phi(x)+\psi(x)$.
So it should be:
$$M((\phi +\psi )(x)) \ \overset{\text{definition of function addition}}{ = } \ M(\phi(x) +\psi(x) )$$

Then we can indeed use that Matrix multiplication is linear to find:
$$M(\phi(x) +\psi(x) ) \ \overset{M \text{ linear }}{ = } \ M(\phi(x)) +M(\psi(x) )
=(M\circ\phi)(x)+(M\circ\psi)(x)=(M(\phi ))(x)+(M(\psi ))(x)$$
(Nod)

2. For $x\in \mathbb{R}^n$ we have the following: $$(M(\lambda \phi ))(x)=M(\lambda \phi )(x) \ \overset{\phi\text{ linear }}{ = } \ M(\lambda \phi(x) ) \ \overset{M \text{ linear }}{ = } \ \lambda M( \phi(x) )=\lambda (M(\phi ))(x)$$

Is everything correct?

Same here:
$$(\lambda\phi)(x) \ \overset{\text{ definition of scalar multiplication with a function}}{ = } \ \lambda(\phi(x))$$
(Thinking)

 

[I like Serena]

We have some new information.
In your latest thread I noticed:

Let $ \sigma_a: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ be the reflection on the straight line through the origin, where $ a $ describes the angle between the straight line and the positive $ x $ axis.

...

Determine the matrix $s_a: = M (\sigma_a)$.

It implies that $M(\phi)$ is the matrix that represents the linear map $\phi: \mathbb R^n\to\mathbb R^m$. (Thinking)

 

[mathmari]

We have some new information.
In your latest thread I noticed:It implies that $M(\phi)$ is the matrix that represents the linear map $\phi: \mathbb R^n\to\mathbb R^m$. (Thinking)

Ah ok.. So can we don't that as in post #8 ? (Wondering)

 

[I like Serena]

Ah ok.. So can we don't that as in post #8 ?

Not quite, because $M(\phi(x))$ is not well defined.
That is, $\phi(x)$ is a vector, and we can't get a matrix from a vector - at least not as intended. (Thinking)

I think it must be as follows.

The matrix of a linear map $\chi:\mathbb R^n\to\mathbb R^m$ is:
$$M(\chi) = \begin{pmatrix}\chi(\mathbf e_1)\cdots \chi(\mathbf e_n)\end{pmatrix}$$
So:
\begin{align*}M(\phi+\psi)
&=\Big[(\phi+\psi)(\mathbf e_1)\,\cdots\, (\phi+\psi)(\mathbf e_n)\Big]\\
&=\Big[(\phi(\mathbf e_1)+\psi(\mathbf e_1))\,\cdots\, (\phi(\mathbf e_n)+\psi(\mathbf e_n))\Big]\\
&=\Big[\phi(\mathbf e_1)\,\cdots\, \phi(\mathbf e_n)\Big]
+\Big[\chi(\mathbf e_1)\,\cdots\, \chi(\mathbf e_n)\Big]\\
&=M(\phi)+M(\psi)\\
\end{align*}
(Thinking)