1. ## Independence, Dependence

HELP! I want to determine if these things are independent/dependent

Don't know how to get the equations for these two:
eg of what I mean by equation

r(a, b) + r2(c, d) + r3(e, f) = 0
ra + r2c + r3e = 0
rb + r2d + r3f = 0

1.
{1, (sinx)^2, cos2x, (cosx)^2} in the F space, real fncts
r1(1) + r2((sinx)^2) + r3(cos2x) + r4((cosx)^2) = 0

from there I don't know how to group the terms so that I can set up a matrix and find out if trivial solutions exist.

2.
Similar problem as # 1 but it's with matrices in the M space of 2x2 matrices
all of the following matrices are 2x2
{ matrix 1, matrix 2, matrix 3}

[ 1 4 ] [ -1 5] [ -1 5]
[ - 3 ] [ -1 5] [ -1 5]

r1Matrix 1 + r2 Matrix 2 + r3 Matrix 3 = 0
then elim to find triv.

I just don't know how to set up the equations never seen examples.

2. Originally Posted by Noxide
HELP! I want to determine if these things are independent/dependent

Don't know how to get the equations for these two:
eg of what I mean by equation

r(a, b) + r2(c, d) + r3(e, f) = 0
ra + r2c + r3e = 0
rb + r2d + r3f = 0

1.
{1, (sinx)^2, cos2x, (cosx)^2} in the F space, real fncts
r1(1) + r2((sinx)^2) + r3(cos2x) + r4((cosx)^2) = 0

$\color{red}\mbox{Do you remember the Trigonometric Pythagoras Theorem }\cos^2x +\sin^2x=1?$

from there I don't know how to group the terms so that I can set up a matrix and find out if trivial solutions exist.

2.
Similar problem as # 1 but it's with matrices in the M space of 2x2 matrices
all of the following matrices are 2x2
{ matrix 1, matrix 2, matrix 3}

[ 1 4 ] [ -1 5] [ -1 5]
[ - 3 ] [ -1 5] [ -1 5]

$\color{red}\mbox{The first one above is not a 2x2 matrix (most probably a typo), but never}$ $\color{red}\mbox{ mind: can you see the 2nd and 3rd matrices are identical?}$

Tonio

r1Matrix 1 + r2 Matrix 2 + r3 Matrix 3 = 0
then elim to find triv.

I just don't know how to set up the equations never seen examples.
.

3. Thanks Tonio

Using trig identities I turned # 1 into:

(cosx)^2 + (sinx)^2, (sinx)^2, (cosx)^2 -(sinx)^2, (cosx)^2

Suppose I then do the following:
r1[(cosx)^2 + (sinx)^2] + r2[(sinx)^2] + r3[(cosx)^2-(sinx)^2] + r4[(cosx)^2] = 0

Is it then correct to do this:
(cosx)^2 Terms: ( r1 + 0r2 + r3 + r4)
(sinx)^2 Terms: ( r1 + r2 - r3 + 0r4)

so that [( r1 + 0r2 + r3 + r4)(cosx)^2] + [( r1 + r2 - r3 + 0r4)(sinx)^2] = 0
ie
[( r1 + 0r2 + r3 + r4)(cosx)^2] =0
[( r1 + r2 - r3 + 0r4)(sinx)^2] = 0
and then create a linear system

[1 0 1 1]
[1 1 -1 0] r = 0

which after row reduction becomes

[1 0 1 1]
[0 1 -2 -1]

so we can then say that there are 2 pivot columns and 2 free columns therefore the presence of free columns suggests that the system has non-trivial solutions and therefore we can say that the vectors are indeed dependent

With #2 I think I may need some help....

I accidently wrote the incorrect matrices while in a rush to get to class. What I meant to write was:

in the M space of 2x2 matrices are these 3 matrices independent or dependent?

[ 1 4 ] [ -1 5] [ 1 13]
[ -1 3 ] [ -1 5] [ 4 7]