Vectors coordinate in a base.

[Randyyy]

Task: Show that the vectors [MATH]{(1,2,1,0),(0,-1,-1,1),(1,1,-1,0),(1,0-3,1)}[/MATH]A: can be used as a base in [MATH]R^4[/MATH]B: Find the coordinate for the vector [MATH](-1,1,1,-1)[/MATH] in this base.

A: I solved by first expressing the system of equation as follows:
[MATH](1): x_1+x_3+x_4=0 [/MATH][MATH](2): 2x_1-1x_2+x_3=0 [/MATH][MATH](3): x_1-x_2-x_3-3x_4=0 [/MATH][MATH](4): x_2+x_4=0 [/MATH]Now I solved it using Gauss elimination and eventually this will yield that the only solution is trivial, [MATH]x_1=x_2=x_3=x_4=0[/MATH] which shows that the system is linearly independent which implies that the four vectors can be used as a base in [MATH]R^4[/MATH].

For B I have no idea what I am supposed to do. My thought is maybe I should find my Transition matrix to convert the coordinates or am I just overcomplicating the problem massively?

 

[Dr.Peterson]

For B I have no idea what I am supposed to do. My thought is maybe I should find my Transition matrix to convert the coordinates or am I just overcomplicating the problem massively?

What would it mean for [MATH](−1,1,1,−1)[/MATH] to have coordinates [MATH](x_1,x_2,x_3,x_4)[/MATH] in that basis?

What matrix might you work with to solve that system of equations? It will look a lot like what you already did ...

 

[Steven G]

You want to write (−1,1,1,−1) as a linear combination of the basis vectors. Then what?

 

[Randyyy]

hmmm, okay I think I get what you are implying Dr.Peterson and Jomo.
To make it easier let's label some things.
Since we have concluded that the 4 vectors make up a base in [MATH]R^4[/MATH], let's call the vectors e with an index. So:
[MATH]e_1=(1,2,1,0), e_2=(0,-1,-1,1), e_3=(1,1,-1,0), e_4=(1,0,-3,1)[/MATH]We can write this as a linear combination as follows: [MATH]e_1x_1+e_2x_2+e_3x_3+e_4x_4=\hat{e} [/MATH] where [MATH]\hat{e}=(-1,1,1,-1) [/MATH]So I get that I need to solve the following system:
[MATH](1): x_1+x_3+x_4=-1[/MATH][MATH](2): 2x_1-x_2+x_3=1[/MATH][MATH](3): x_1-x_2-x_3-3x_4=1[/MATH][MATH](4):x_2+x_4=-1[/MATH]Using Gaussian elimination again I eventually end up with [MATH](x_1,x_2,x_3,x_4)=(1, -4, -5, 3)[/MATH] which is inline with what wolfram gets the system to be. So the vector [MATH]\hat{e}[/MATH] has got the coordinates [MATH](1, -4, -5, 3)[/MATH] in our base in [MATH]R^4[/MATH]being spanned up by [MATH]e_1,e_2,e_3,e_4[/MATH].




I do think that my naming and labelling probably is a bit confusing but I wasn´t too sure how to label everything so that it would make sense.

 

[Steven G]

Instead of using e_i you could have used any other variable like v_i. Personally I like the variable harvey (I've improved it used to be Subhotosh)

 

[Randyyy]

Thank you for the help Jomo and Dr.Peterson!