# Matrices and linear maps.

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• Nov 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)
• Nov 23rd 2011, 08:43 AM
Deveno
you need to prove that $\displaystyle f(x_1+y_1,x_2+y_2) = f(x_1,x_2) + f(y_1,y_2)$ and $\displaystyle f(ax_1,ax_2) = a(f(x_1,x_2))$, for starters.
• Nov 23rd 2011, 08:55 AM
flexx
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.
• Nov 23rd 2011, 10:17 AM
Deveno
show me what you have so far, and where you run into trouble....
• Nov 23rd 2011, 10:42 AM
flexx
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?
• Nov 23rd 2011, 11:56 AM
Siron
Given is the map $\displaystyle f: \mathbb{R}^2 \to \mathbb{R}^2: \binom{x_1}{x_2}\to \binom{x_1}{\frac{x_2}{3}}$

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

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

Can you continue?
• Nov 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)
• Nov 23rd 2011, 12:29 PM
Siron
Re: Matrices and linear maps.
No problem. To finish (1):
$\displaystyle f\left[\binom{a_1+b_1}{a_2+b_2}\right]=\binom{a_1+b_1}{\frac{a_2+b_2}{3}}=\binom{a_1+b_1 }{\frac{a_2}{3}+\frac{b_2}{3}}=\binom{a_1}{\frac{a _2}{3}}+\binom{b_1}{\frac{b_2}{3}}=f\left[\binom{a_1}{a_2}\right]+f\left[\binom{b_1}{b_2}\right]$

Do you understand this?

Try to prove (2)
• Nov 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?
• Nov 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:
$\displaystyle \forall \binom{a_1}{a_2} \in \mathbb{R}^2,\forall \lambda \in \mathbb{R}: f\left[\lambda \cdot \binom{a_1}{a_2}\right]=\lambda \cdot f\left[\binom{a_1}{a_2}\right]$
That means
$\displaystyle f\left[\lambda \cdot \binom{a_1}{a_2}\right]=f\left[\binom{\lambda \cdot a_1}{\lambda \cdot a_2}\right]= ...$

Can you continue?
• Nov 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:

$\displaystyle f(x_1,x_2) = (x_1,\frac{x_2}{3})$ and $\displaystyle f:\binom{x_1}{x_2} \to \binom{x_1}{\frac{x_2}{3}}$ are just notational differences, one can either represent a vector in $\displaystyle \mathbb{R}^2$ 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 $\displaystyle \mathbb{R}^2$).

to create a "scalar multiple" of $\displaystyle (x_1,x_2)$, (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".
• Nov 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!
• Nov 23rd 2011, 01:49 PM
Siron
Re: Matrices and linear maps.
$\displaystyle \binom{a_1}{a_2},\binom{b_1}{b_2}$ and $\displaystyle \binom{x_1}{x_2}$ are just vectors in $\displaystyle \mathbb{R}^2$, 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).
• Nov 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?
• Nov 23rd 2011, 05:28 PM
Deveno
Re: Matrices and linear maps.
we are not multiplying f and lambda. what we need to prove is:

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

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:

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

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

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

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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