Explaining General Linear Position & GL Group

[jeff1evesque]

Statement:The following definition was taken from wikipedia:
The general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible. The name is because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position, and matrices in the general linear group take points in general linear position to points in general linear position.

To be more precise, it is necessary to specify what kind of objects may appear in the entries of the matrix. For example, the general linear group over R (the set of real numbers) is the group of n×n invertible matrices of real numbers, and is denoted by $$GL_{n}(R)$$ or $$GL(n, R).$$

Question:
Can someone explain to me what a general linear position is.

Also what is meant by the following two statements (taken from the statement above):
(i.)

The name is because the columns of an invertible matrix are linearly independent, hence the vectors/points they define are in general linear position.

(ii.)

...and matrices in the general linear group take points in general linear position to points in general linear position.

Thanks,JL

 

[Dick]

Concentrate on "linearly independent". They appear to be using "general position" as a vague synonym for linearly independent. It doesn't mean anything different.

 

[hysteria]

Hey can i get your thoughts on this one? i can't seem to figure out, its an internal resistance and electromotive force problems

1) a complete circuit consists of a 12.0 V battery with a 4.50 ohm resistor and switch. the internal resistance of the battery is 0.30 ohms when switch is open. what does the voltmeter read when placed:

a. across the terminal of the battery when the switch is open
b. across the resistor when the switch is open
c. across the terminal of the battery when the switch is closed
d. across the resistor when the switch is closed

2) when the switch S is open, the voltmeter V reads 2.0 V. when the switch is closed, the voltmeter reading drops to 1.50 V and the ammeter reads 1.20 A. find the emf (electromotive force) and the internal resistance of the battery. assume that the two meters are ideal so they don't affect the circuit.

 

[Dick]

Maybe you can't figure it out, but you can try. And this has nothing to do with GL(n,R). You really should post a new thread and show SOME attempt at solving it before expecting any help.

 

[jeff1evesque]

I think I get the point, but here it is anyways (I find it hard to solve circuit problems also- if i made a mistake, forgive me):

Hey can i get your thoughts on this one? i can't seem to figure out, its an internal resistance and electromotive force problems

1) a complete circuit consists of a 12.0 V battery with a 4.50 ohm resistor and switch. the internal resistance of the battery is 0.30 ohms when switch is open. what does the voltmeter read when placed:

a. across the terminal of the battery when the switch is open
b. across the resistor when the switch is open
c. across the terminal of the battery when the switch is closed
d. across the resistor when the switch is closed

When the circuit is closed we have the following:
Current across the closed circuit:
$$I = \frac{V}{R} = \frac{12}{4.5} = 2.67 amps$$ by ohms law.
Also, the voltage across the resistor is the same as the voltage across the battery, which can be verified by Ohms law $$V = I \cdot R.$$

2) when the switch S is open, the voltmeter V reads 2.0 V. when the switch is closed, the voltmeter reading drops to 1.50 V and the ammeter reads 1.20 A. find the emf (electromotive force) and the internal resistance of the battery. assume that the two meters are ideal so they don't affect the circuit.

I am not sure about this. Is this a parallel circuit, and where is the voltmeter hooked up to?

Thank you,JL