1. ## Dependent/Combination Vectors

1) Let V be a vector space and $\displaystyle a,b,c,d \in V$. Show that the following vectors are linearly dependent.

$\displaystyle v_1 = a + b + c -44d$
$\displaystyle v_2 = a - 3b + 6c$
$\displaystyle v_3 = b + d$
$\displaystyle v_4 = 9a + 7c + 5d$
$\displaystyle v_5 = 23b - c - d$

So is it...

[ 1 1 0 9 0 ][a] = [0]
[ 1 -3 1 0 23][b] = [0]
[ 1 6 0 7 -1][c] = [0]
[-44 0 1 5 -1][d] = [0]

?

Will that not give a=b=c=d=0, which is not really a lot of use because could that not just be applied to any combination of vectors? I thought about using Gaussian elimination but again thats not gonna work and i dont know how to use it on an m x n (m not equal to n{whats the latex for this?}) matrix. So how do you do it?

2) Write w as the linear combination of the vectors $\displaystyle v_1, v_2, v_3$
$\displaystyle w =(2,1,1), v_1 = (1,5,1), v_2 = (0,9,1), v_3 = (-3,3,1)$

Now i can see almost straight away that $\displaystyle w = v_3 + v_2 - v_1$ but im not sure if the correct method to actually solve these. Is there one? Guassian Elimination?

2. actually ive solved part 2 so its just 1 now.

1) Let V be a vector space and $\displaystyle a,b,c,d \in V$. Show that the following vectors are linearly dependent.

$\displaystyle v_1 = a + b + c -44d$
$\displaystyle v_2 = a - 3b + 6c$
$\displaystyle v_3 = b + d$
$\displaystyle v_4 = 9a + 7c + 5d$
$\displaystyle v_5 = 23b - c - d$

So is it...

[ 1 1 0 9 0 ][a] = [0]
[ 1 -3 1 0 23][b] = [0]
[ 1 6 0 7 -1][c] = [0]
[-44 0 1 5 -1][d] = [0]

?

Will that not give a=b=c=d=0, Mr F says: Yes. Therefore the set is dependent.

which is not really a lot of use Mr F says: yes it is.

because could that not just be applied to any combination of vectors? Mr F says: Of these vectors, yes. Is it really that surprising that in a vector space V of dimension four, five vectors from V will always be linearly dependent?

I thought about using Gaussian elimination but again thats not gonna work and i dont know how to use it on an m x n (m not equal to n{whats the latex for this?}) matrix. So how do you do it?

2) Write w as the linear combination of the vectors $\displaystyle v_1, v_2, v_3$
$\displaystyle w =(2,1,1), v_1 = (1,5,1), v_2 = (0,9,1), v_3 = (-3,3,1)$

Now i can see almost straight away that $\displaystyle w = v_3 + v_2 - v_1$ but im not sure if the correct method to actually solve these. Is there one? Guassian Elimination?
..