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Let $$S = \{T:\mathbb{R}^3\to\mathbb{R}^3:T\text{ is a linear transformation with }T(1,0,1)=(1,2,3),T(1,2,3)=(1,0,1)\}$$ Then S is:

(a) a singleton set

(b) a finite set containing more than one element

(c) a countably finite set

(d) an uncountable set

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What have you done, what have you tried, ideas, insights, self work...? – DonAntonio Jan 8 '13 at 11:18
pls tell me the true option – aman Jan 8 '13 at 11:21
First, you already have an answer, given without you making a minimal effort to show that you've taken some time to think and solve the problem, which seems to be homework. This is not how SE is supposed to work. – DonAntonio Jan 8 '13 at 11:25
up vote 0 down vote accepted

It's an uncountable set because you can choose another vector $v$ to form $\{(1,0,1),(1,2,3),v\}$ a basis and so for each $x \in \mathbb{R}$ there exist, in fact unique, $T \in S$ such than $T(v) = (x,0,0)$.

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thanks a lot.... – aman Jan 8 '13 at 11:34

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