Math homework

[tautis154]

Hello, I've been struggling on these problems that are in the picture and I would like some help. Thank you

 

[raquelc]

For the top bit, you need to link each of the coordinates.
u=1x+2y
v=1x+3y
w=2x+3y

Multiplying the first one and the 2nd by 2:

2u=2x+4y
2v=2x+6y
w=2x+3y

1st - 2nd: 2u-2v=-2y
1st - 3rd: 2u-w=y

Multiplying this last one by 2
2u-2v=-2y
4u-2w=2y

And adding them together
6u-2v-2w=0

you can simplify dividing by 2
3u-v-w=0

 

[raquelc]

For a)

the general vector for S will be (x+y , y , x+2y)

We need to prove that exists the 0 vector, it is closed under addition, and closed under multiplication by an escalar

When x=y=0, then we have the null vector (0,0,0)

(x+y , y , x+2y)+(a+b , b , a+2b) = ( (x+a)+(y+b) , y+b , (x+a)+2(y+b))

k*(x+y , y , x+2y)= (kx+ky , ky , kx+2ky)

 

[raquelc]

b)

(x+y, y, x+2y)=(2m , m+3 , 0)

x+y=2m
y=m+3
x+2y=0

From the 3rd x=-2y
Subbing on the first
-2y+y=2m
-y=2m

looking that with the 2nd and adding them together

y=m+3
-y=2m

0=3m+3
m=-1

 

[Deleted member 4993]

b)

(x+y, y, x+2y)=(2m , m+3 , 0)

x+y=2m
y=m+3
x+2y=0

From the 3rd x=-2y
Subbing on the first
-2y+y=2m
-y=2m

looking that with the 2nd and adding them together

y=m+3
-y=2m

0=3m+3
m=-1

Please DO NOT post complete solution when the OP has shown NO EFFORT.