Matrices and linear maps.

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November 23rd 2011, 08:06 AM
flexx

Matrices and linear maps.

f: R2 --> R2

X1 --> X1
X2 --> 1/3X2

There's only meant to be 1 arrow between the two matrices.

Prove that f is a linear map using additivity and homogeneity.
Determine the matrix which represents this linear map.

I have no idea what to do :s.

Any help would be greatly appreciated!

Thank you! (Clapping)
November 23rd 2011, 08:43 AM
Deveno

Re: Matrices help please!!?

you need to prove that $f(x_1+y_1,x_2+y_2) = f(x_1,x_2) + f(y_1,y_2)$ and $f(ax_1,ax_2) = a(f(x_1,x_2))$ , for starters.
November 23rd 2011, 08:55 AM
flexx

Re: Matrices help please!!?

Ok, thanks. Any more suggestions after that?

I'm an exchange student and i cannot follow the lectures in the language set by the university, i'd appreciate if you could go through this example maybe so i could attempt other questions similar.
November 23rd 2011, 10:17 AM
Deveno

Re: Matrices help please!!?

show me what you have so far, and where you run into trouble....
November 23rd 2011, 10:42 AM
flexx

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Re: Matrices help please!!?

I know that to prove additivity, you need to show that for any real x_1, x_2, y_1, y_2 you have f{x_1\choose x_2} + f{y_1\choose y_2} = f{x_1+y_1\choose x_2+y_2}

and to prove homogeneity, you need to show that for any real x_1, x_2, alpha you have f{\alpha x_1\choose \alpha x_2} = \alpha f{x_1\choose x_2}, i need to evaluate the left and right hand sides of the equation and show that i get the same answer.

However, i am not sure how to evaluate the left and right hand sides of the equations? could you clarify this in detail or actually solve it please?
November 23rd 2011, 11:56 AM
Siron

Re: Matrices help please!!?

Given is the map $f: \mathbb{R}^2 \to \mathbb{R}^2: \binom{x_1}{x_2}\to \binom{x_1}{\frac{x_2}{3}}$

To prove: $f$ is a linear map.
You have to show that:
(1) $\forall \binom{a_1}{a_2}, \binom{b_1}{b_2} \in \mathbb{R}^2: f\left[\binom{a_1}{a_2}+\binom{b_1}{b_2}\right]=f\left[\binom{a_1}{a_2}\right]+f\left[\binom{b_1}{b_2}\right]$
(2) $\forall \binom{a_1}{a_2}\in \mathbb{R}^2, \forall \lambda \in \mathbb{R}: f\left[\lambda \cdot \binom{a_1}{a_2}\right]=\lambda \cdot f\left[\binom{a_1}{a_2}\right]$

For example to prove (1):
$f\left[\binom{a_1}{a_2}+\binom{b_1}{b_2}\right]=f\left[\binom{a_1+b_1}{a_2+b_2}\right]=...$

Can you continue?
November 23rd 2011, 12:20 PM
flexx

Re: Matrices and linear maps.

I'm sorry but i am really not sure how to continue, i've only just started this topic and cannot find anything related on the web and my course is not in English -_-

Is it possible to continue please? (Itwasntme)
November 23rd 2011, 12:29 PM
Siron

Re: Matrices and linear maps.

No problem. To finish (1):
$f\left[\binom{a_1+b_1}{a_2+b_2}\right]=\binom{a_1+b_1}{\frac{a_2+b_2}{3}}=\binom{a_1+b_1 }{\frac{a_2}{3}+\frac{b_2}{3}}=\binom{a_1}{\frac{a _2}{3}}+\binom{b_1}{\frac{b_2}{3}}=f\left[\binom{a_1}{a_2}\right]+f\left[\binom{b_1}{b_2}\right]$

Do you understand this?

Try to prove (2)
November 23rd 2011, 12:45 PM
flexx

Re: Matrices and linear maps.

Thank you so much!, i was able to follow (1), for (2): do i replace x1 and x2 by a1 and a2/3?
November 23rd 2011, 01:05 PM
Siron

Re: Matrices and linear maps.

Can you be more specific about what you mean? We have to prove that:
$\forall \binom{a_1}{a_2} \in \mathbb{R}^2,\forall \lambda \in \mathbb{R}: f\left[\lambda \cdot \binom{a_1}{a_2}\right]=\lambda \cdot f\left[\binom{a_1}{a_2}\right]$
That means
$f\left[\lambda \cdot \binom{a_1}{a_2}\right]=f\left[\binom{\lambda \cdot a_1}{\lambda \cdot a_2}\right]= ...$

Can you continue?
November 23rd 2011, 01:31 PM
Deveno

Re: Matrices and linear maps.

the idea of a linear map, is that it preserves vector sums, and scalar multiples. note that:

$f(x_1,x_2) = (x_1,\frac{x_2}{3})$ and $f:\binom{x_1}{x_2} \to \binom{x_1}{\frac{x_2}{3}}$ are just notational differences, one can either represent a vector in $\mathbb{R}^2$ horizontally or vertically, either way we need "two coordinates" (and this number-of-coordinates thing, is actually important, it's the main distinguishing feature of $\mathbb{R}^2$ ).

to create a "scalar multiple" of $(x_1,x_2)$ , (no matter which way you write it), you multiply each coordinate by the scalar (which in this case means a real number, rather than a 2-vector).

so we should (if f is linear), get the same thing if we "do f first" and then multiply by a scalar, or if we multply by a scalar first, and then do f to the "scaled vector".

just as additivity means we can do f to each part of a sum separately, and then add, or we can add first, and then do f. some books phrase this as:

"f respects vector addition and scalar multiplication".
November 23rd 2011, 01:43 PM
flexx

Re: Matrices and linear maps.

Thanks for the replies, i am just not so sure since now we are only dealing with a1 and a2. Would it be ok to show me?

I really appreciate all the help :) i have tons of examples to go through after these!
November 23rd 2011, 01:49 PM
Siron

Re: Matrices and linear maps.

$\binom{a_1}{a_2},\binom{b_1}{b_2}$ and $\binom{x_1}{x_2}$ are just vectors in $\mathbb{R}^2$ , actually it doesn't matter how you call them, you just have to realise we're working in a 2-dimensional vector space so there are two components.

I think you can finish the proof now? (show us)

To determine the matrix which represents the linear map, have you any ideas about that? (think about bases of vector spaces).
November 23rd 2011, 02:14 PM
flexx

Re: Matrices and linear maps.

Continuing from where you left off:

f . lambda ( [ x1, x2/3] )
=>

f . lambda [ x1 , x2/3]

Is this correct?
November 23rd 2011, 05:28 PM
Deveno

Re: Matrices and linear maps.

we are not multiplying f and lambda. what we need to prove is:

$f\left(\lambda\begin{bmatrix}x_1\\x_2\end{bmatrix} \right) = \lambda f\left(\begin{bmatrix}x_1\\x_2\end{bmatrix}\right)$ .

f is a function that takes as its input 2 variables, and outputs two values. to prove the equality above, we need to calculate what f(___) is, using the definition of f.

so first we calculate:

$\lambda\begin{bmatrix}x_1\\x_2\end{bmatrix} = \begin{bmatrix}\lambda x_1\\ \lambda x_2\end{bmatrix}$

then we use the definition of f, to calcuate the two values of f(λx):

$f\left(\begin{bmatrix}\lambda x_1\\ \lambda x_2\end{bmatrix}\right) = \begin{bmatrix}\lambda x_1\\ \frac{\lambda x_2}{3}\end{bmatrix}$

now start on the other side with f(x) = f(x1,x2), calculate it, and then multiply each coordinate by λ, what do you get?

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