- #1

vcb003104

- 19

- 0

## Homework Statement

let r be an element of R

... 1.... 1 ......r^2.....3 + 2r

u =( 1 )...v = ( 4 )...w = (1 )...b = ( 5 + 12r)

...0.....1......r^2 ...... 2r

(sorry don't know how to type matrices)

1. For which values of r is the set {u, v, w} linearly independent?

2. For which values of r is the vector b a linear combination of u. v, w?

3. For which of these values of r can b be written as a linear combination of u, v and w in more than one way

## Homework Equations

(the matrices)

## The Attempt at a Solution

So for 1, I reduced the matrix ( u v w ) to become something like this:

_1_1_r^2

(_1_4_1)

0_1_r^2

__1_1_r^2

→ (0_1_r^2)

__0_0_1-4r^2

so for it to be linearly independent 1 - [itex] 4r^2 [/itex] =/= 0

so r =/= [itex]\pm 1/2[/itex]

for part b)

We want something like:

c1(u) + c2(v) + c3(w) = b

I reduced everything and got:

c1 = 3

c2 = 2r - [itex] r^2 [/itex] (2/(1 - 2r))

c3 = [itex]\frac{2(1 + 2r)}{(1 + 2r)(1 - 2r)}[/itex]

Is it alright to say that for it to be a linear comb. r can't = [itex]\pm1/2[/itex]? (Is it correct that I didn't cancel out the 1 + 2r ?)

and I don't really get part C