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This is where I am stuck. I studied ordered basis and coordinates vector previous to this.

of course I studied vector space, basis, linear... etc too,

However I can't understand just this part. (maybe this whole part)

Especially

this one which says [[T(b1)]]c....[[T(bn)]]c be a columns of matrix.

Can any one please explain me how this works? I've stuck at here too long.