Question

I don't understand the coordinates with respect to a basis? Find the (w)_B and [w]_B for R^3 given that w = (2, -1 ,3); v_1=(1,0,0), v_2=(2,2,0) and v_3=(3,3,3)

Answer 1

What I did was dot product w with each of the vectors and got 2,2,12 and assumed that (w)_B was (2,212)

Answer 2

2,2,12*

Answer 3

Are the \(v_1, v_2, v_3\) are the basis of the space \(\mathcal{B}\)?

Answer 4

Yeah B={v_1,v_2,v_3} for R^3

Answer 5

Never understand why people don't just use orthogonal bases.

Answer 6

Estudier, that comes later on in linear algebra. Any linearly independent set can be transformed into an orthogonal basis via the Gram Schmidt process. :)

Answer 7

Okay well I have to learn about coordinates wrt to basis b/c it's part of my course.. So I still don't understand in my textbook it states that (w)_B=(w dotted with v_1),(w dotted with v_2).. etc

Answer 8

Estudier, that comes later on in linear algebra. Any linearly independent set can be transformed into an orthogonal basis via the Gram Schmidt process. :)

Sure, but what is the use of any other sort?

Answer 9

I did that and got 2,2, 12 but the answer is 3,-2,1

Answer 10

Come on Joe, sort it....:-)

Answer 11

im going out to eat in like 2 mins >.<

Answer 12

celebratory "Summer semester is finally over" lunch lol

Answer 13

Awe well when you get back? I'm just studying.. Linear Algebra is my last final!

Answer 14

Well, as best as I can understand it, what u are saying is right.

U have (2,-1,3) in V(R3) spanned by v1,v,2,v3 (orthonormal, right?)

So u dot'em and get 2,2,12

Answer 15

Yeah that's how I understood it but the answer in the back is 3,-2,1... Don't entirely get this last chapter, coordinates are confusing so I'm just going by the formulas given so hopefully it's a mistake

Answer 16

Maybe Joe will think of something when he gets back..

Answer 17

Oh nvm I got it, they just expressed w as a linear combination of the three vectors then put them into a matrix and row reduced. The system has three unique solution 3, -2, 1. So w=3(1,0,0) + -2(2,2,2)+1(3,3,3). I just don't understand why the equation I used earlier didn't work.

Answer 18

For what it is worth, your basis v1,v2,v3 is not orthogonal (nor unit length). So the projection onto v1 and v2 is, in some sense, double counted.
contrast the orthogonal basis (1,0), (0,1) with (1,0), (1,1), and find the coords of point (1,1).

Answer 19

For what it is worth, your basis v1,v2,v3 is not orthogonal (nor unit length)

Yup, that would be a bit of a problem...