Linear Algebra Q (1 Viewer)

[VBN2470]

Need someone to clarify what the first part means (+ method):



Thanks

 

[mreditor16]

I wish I could help :/

 

[erckle999]

Need someone to clarify what the first part means (+ method):

View attachment 32021

Thanks

You've probably figured this out by now, but anyways:
If you could show that three vectors given in the polynomial space B={1-t, 1-t^2, 1+t-t^2} formed a basis for the polynomial space, then this would mean any vector in the polynomial space could be written as a unique sum of the vectors in B. Therefore, T applied to any vector in the domain could map to only one vector in the co-domain.

Conversely, if B was not a basis, then some v in the domain could be written as a linear combination of B in more than one (actually infinite) ways. Thus, T(v) could be calculated in more than one (infinite) ways so T would not be determined uniquely.

To do the second part, if you can write the standard basis of the domain (e1, e2, e3) in terms of B, you could then find T(e1), T(e2) and T(e3). Since the question gives you T with respect to to B in the domain and the standard basis in the codomain, you would be done. The desired matrix T would have T(e1), T(e2), T(e3) as columns.

Good luck.
(PS how do you put Latex into these posts?)

 

[VBN2470]

Yep I managed to figure it out before, but thanks anyways To use LaTeX, you need to insert [tex.] [/tex.] command (without the dots) and insert your text/mathematics between them. Use $$ signs too. Use this guide as a starting point: http://community.boredofstudies.org/14/mathematics-extension-2/234259/short-guide-latex.html

Hope it helps