Linear transformation question

Oct 2012
15
0
HK
I've got a question like this:

Let V be a finite dimensional vector space with dimV >= 2 . Show
that there exists a linear transformation T:V to V such that T not equal to zero but
T^2 = 0 .

I have figured out the "T not equal to zero but T^2 = 0 ." part already, rankT>=1, dimNulT>=rankT>=1, satisfies with dimV>=2 by rank theorem, but how to present it nicely?
 
Last edited:

Deveno

MHF Hall of Honor
Mar 2011
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Tejas
pick a basis B = {b1,...,bn} for V.

thus we can write any v in V as v = c1b1+...+cnbn.

define T(c1b1+...+cnbn) = cnb1.

T is non-zero because T(bn) = b1 ≠ 0, since b1 is a basis vector.

but T2(v) = T(T(v)) = T(T(c1b1+...+cnbn)) = T(cnb1) = 0

(because cnb1 = cnb1 + 0b2+....+0bn).

basically we are using the linear transformation that has the matrix (relative to the basis B):

\(\displaystyle \begin{bmatrix}0&0&\dots&0&1\\0&0&\dots&0&0\\ \vdots&\vdots&\ddots&\vdots&\vdots\\0&0&\dots&0&0 \end{bmatrix}\)
 
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