help please! vectors!?

Question

if u=(1,0,2), v=(0,3,-1) and w=(1,1,2) , how do i express (-1,0,-3) as a linear combination of u, v and w.

[u, v and w are meant to be bold letters but i dont know how to do that on this? so they're vectors, yes?]

also can all vectors in R³ be expressed as linear combinations of these three vectors?

- - - - - - - - - - -

if u=(1,2,1) and v=(-1,1,2) and w=(5,4,-1), use the following fact and no row operations to solve 3u-2v-w=0 :

1 -1 x 5
2 1 y = 4
1 2 -1

[incase thats unclear above is three matrices.]

thank you

the matrices really didnt come out clear.

A=
1 -1
2 1
1 2

B=
x
y

C=
5
4
-1

and AB=C

thanks.

Answer 1

Given vectors u,v, and w

u = <1,0,2>
v= <0,3,-1>
w= <1,1,2>

Express <-1,0,-3> as a linear combination of u, v and w.

a(i + 2k) + b(3j - k) + c(i + j + 2k) = - i - 3k

a + c = -1
3b + c = 0
2a - b + 2c = -3

Solving three equations in three unknowns we get
a = 2
b = 1
c = -3

So

3u + v - 3w = <-1,0,-3>
_______________________

Can all vectors in R³ be expressed as linear combinations of these three vectors?

Let's look at the the same equations again except that they are all equal to zero. If the only solution is the trivial one, namely that a = b = c = 0. Then the vectors span R³ and some linear combination of them can represent any vector in R³.

a + c = 0
3b + c = 0
2a - b + 2c = 0

Indeed, when we solve this series of equations the trivial solution is the only solution so the vectors u,v, and w span R³.
___________________

The second question is unclear.

Answer 2

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Answer 3

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