Exercise of algebra?

Posted on 2015-09-01 06:13:00 in Math by PATRIOT

Hi!

Can please help me with this exercise?
It says:
1)Given S={(x?,x?,x?) ? R^3 : x? + x? -2x? = 0}.
a)Prove that S is a subspace.
b)For each of the matrices A shown, check if possible find two bases B and B' of subspace S -
so C(over BB')= A.
If possible, find them and show these bases. Otherwise unable to find them, explain why.
(i)A=
[2..-3]
[1..-2]

(ii)A=
[4..-6]
[-2..3]

2)Given the linear transformation T:R^4 --> P?[R] such that:
Ker(T) = {(x?,x?,x?,x?) ? R^4 : 2x? - x? + x? = 0, 2x? - x? = 0}, T(0,1,0,1) = -2x^2 + x and T(0,0,0,1) = x^2 - 2.
a)Check if X - 4 belongs to the subspace of Im(T).
b)Calculate T(1,1,5,0).

Thank you!

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