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Let $T$ be a linear transformation on the real vector space $R^n$ over $R$ such that $T ^2 =μT$ for same $μ∈R$ . Then which of the following is/are true?

  1. $||Tx|| = |μ| || x||$ for all $x ∈R^n$

  2. If $||Tx|| = || x|| $for some non zero vector $x ∈R^n$, then $μ=±1$

  3. $T= μI$ where $I$ is the identity transformation on$R^n$

  4. If $||Tx ||>x$ for a non zero vector $x \in R^n$, then $T$ is necessarily singular.

I am completely stuck on it. Can anybody help me please?

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1 Answer 1

Hint: Consider $T=\begin{pmatrix}\sqrt{2}&0\\0&0\end{pmatrix}$ for items 1-3. In each of item 1 and item 2, find a suitable $x$ to refute the statement. For item 4, consider $T=2I$.

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can u elaborate please.thanks –  damini Dec 13 '12 at 3:57
    
i am slightly confuse how to calculate the norm here –  damini Dec 13 '12 at 3:57
    
Does your textbook specify the norm? If not, we usually take the Euclidean 2-norm, i.e. $\|(x_1,\ldots,x_n)^T\|=\sqrt{x_1^2+\ldots+x_n^2}$. In our case, $\|(x,y)^T\|=\sqrt{x^2+y^2}$, i.e. calculate the length of a vector using Pythagoras theorem. –  user1551 Dec 13 '12 at 4:08

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