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Let $V$ be the vector space of all real polynomials of degree $\leq 10$. Let $Tp(x)=p'(x)$ for $p\in V$ be a linear transformation from $V$ to $V$. Consider the basis $\{1,x,x^2,\cdots x^{10}\}$ of $V$. Let $A$ be the matrix of $T$ with respect to this basis. Then which of the following is correct?

  1. Trace $A=1$
  2. There is no $m\in \mathbb{N}$ Such that $A^m=0$.
  3. $\det{A}=0$.
  4. $A$ has a nonzero eigenvalue. 
in Linear Algebra by Expert (2.4k points) | 33 views

1 Answer

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Note that  the matrix of given linear transformation is

$$A= \begin{bmatrix}  0 & 1 & 0 &0& \cdots & 0 \\ 0 & 0  & 2 &  0 & \cdots & 0\\ 0 & 0  & 0 & 3 & \cdots & 0\\ \vdots& \vdots & \vdots&\ddots&\ddots&\vdots\\0 & 0 & 0  & \cdots & 0& 10\\ 0 & 0 & 0 & 0 & \cdots & 0 \end{bmatrix}$$

This is an upper triangular matrix with diagonal entries are zero. Thus determinant is equal to zero. Hence, option $3$ is correct.

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