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October 02, 2025, 01:30:30 am
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Author Topic: Linear transformation  (Read 2174 times)  Share 

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Linear transformation
« on: February 17, 2010, 01:52:09 am »
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Just a quickie...

Show that the mapping T defined by is not linear.

Does the notation mean the same thing as ?

And so would this be the way to go about it?



Thanks!
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Mao

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Re: Linear transformation
« Reply #1 on: February 17, 2010, 02:02:13 am »
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Yes. The notation is essentially the same, just different ways of representing 'vectors' (for lack of a better expression).
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Re: Linear transformation
« Reply #2 on: February 17, 2010, 02:05:46 am »
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thx mao!
also the column vectors friggin pwn the ( ) vectors
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mark_alec

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Re: Linear transformation
« Reply #3 on: February 17, 2010, 02:07:41 am »
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Easier way: show that (0,0) does not map to (0,0,0).
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Re: Linear transformation
« Reply #4 on: February 17, 2010, 02:10:12 am »
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Quote from: mark_alec on February 17, 2010, 02:07:41 am
Easier way: show that (0,0) does not map to (0,0,0).

Does that always prove it is a linear transformation? How come?
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Mao

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Re: Linear transformation
« Reply #5 on: February 17, 2010, 02:19:36 am »
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Quote from: /0 on February 17, 2010, 02:10:12 am
Quote from: mark_alec on February 17, 2010, 02:07:41 am
Easier way: show that (0,0) does not map to (0,0,0).

Does that always prove it is a linear transformation? How come?

If

Then

Cbf trying to see if this always proves if something is a linear transformation, but it definitely disproves it.

EDIT: it's not 'iff', T can be a non-linear transformation and still satisfies , such as
« Last Edit: February 17, 2010, 02:24:45 am by Mao »
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Re: Linear transformation
« Reply #6 on: February 17, 2010, 02:24:18 am »
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Ah ok, thanks
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mark_alec

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Re: Linear transformation
« Reply #7 on: February 17, 2010, 02:37:42 am »
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Quote from: /0 on February 17, 2010, 02:10:12 am
Quote from: mark_alec on February 17, 2010, 02:07:41 am
Easier way: show that (0,0) does not map to (0,0,0).
Does that always prove it is a linear transformation? How come?
One of the properties of a linear transformation that must be satisfied is that the 0 maps onto 0, so it is often a trivial way of showing that something is *not* a linear transformation.
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QuantumJG

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Re: Linear transformation
« Reply #8 on: February 17, 2010, 12:52:15 pm »
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Quote from: /0 on February 17, 2010, 02:10:12 am
Quote from: mark_alec on February 17, 2010, 02:07:41 am
Easier way: show that (0,0) does not map to (0,0,0).

Does that always prove it is a linear transformation? How come?

I don't exactly know why it is so, but I know I'm REALLY rusty with linear algebra.

Quote from: Mao on February 17, 2010, 02:19:36 am
Quote from: /0 on February 17, 2010, 02:10:12 am
Quote from: mark_alec on February 17, 2010, 02:07:41 am
Easier way: show that (0,0) does not map to (0,0,0).

Does that always prove it is a linear transformation? How come?

If

Then

Cbf trying to see if this always proves if something is a linear transformation, but it definitely disproves it.

EDIT: it's not 'iff', T can be a non-linear transformation and still satisfies , such as

As what mao said, it doesn't always tell us the transformation is linear, but if this condition is not satisfied then no matter what the transformation is not linear.
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mark_alec

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Re: Linear transformation
« Reply #9 on: February 17, 2010, 03:10:01 pm »
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If is a linear transformation then the following are true:
1.
2.

The zero vector mapping to the zero vector is a consequence of this. If you can find a counter-example to any of these properties then is not a linear transformation.
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