Question

Find the basis for the given subspace of R^3, and state its dimension for the plane 3x-4y+5z=0.

Answer 1

what are 2 linearly independant vectors in the plane?

Answer 2

I'm having trouble understanding that concept.

Answer 3

the basis for a plane consists of 2 vectors by which any of the vectors in the plane can be represented as a linear combination of those 2

Answer 4

for example, in the standard xy plane, 0,1 and 1,0 form the basis from which all the other vectors can be created

if x = 0,1 and y = 1,0

then all vectors can be represented as some v = kx + cy for some constants c and k

Answer 5

we are given a plane, so we need 2 vectors in it that are not a scalar of each other so that they give us 2 distinct directions to play with

Answer 6

spose x=0, and y=0, what is z?

Answer 7

0

Answer 8

and we really dont want the vector 0,0,0 to be part of the basis ..

lets try some other values

lets x=1, y=0, what is z?

Answer 9

can be any number? i will choose 2.

Answer 10

well, when we define x and y, z is predetermined for us

solveing for z: z = (4y-3x)/5

if x=1 and y=0, s=-3/5

Answer 11

**z=-3/5

Answer 12

if x=0 and y=1, z=4/5

Answer 13

but these arent what we like to see vectors as, fractions are just ugly i spose so

v1 = 1,0,-3/5
v2 = 0,1,4/5

or in a more pretty view

v1 = 5,0,-3
v2 = 0,5,4

unless of course they want a unit vectors

Answer 14

or orthogonal vectors ...

Answer 15

is this the only basis vectors we can find for it? no

Answer 16

wait, if you are solving for z, shouldn't that 5 be negative?

Answer 17

why are you thinking that? tell me yor reason