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Difference between mapping R3 to R2 and the reverse

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1. Let V be a vector space of odd dimension (greater than 1) over the real field R. Show that any linear operator on V has a proper invariant subspace other than {0}.

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This solution shows how to prove that any linear operator on V has a proper invariant subspace other than {0}.

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Please see the attachment.

Suppose is a linear operator on , where is a vector space over the real field and is odd. By the definition of a linear operator, for any and , we have and .
Now I show that has a proper subspace other than .
Since is a vector space, it has a basis . Each can be ...

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