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Let \(V\) be a vector space over a field \(F\), with dimension \(\dim(V) = 3\). \(T: V \to V\) is a linear transformation. There exists a vector \(v \in V\) such that \(T^2v \neq 0\) and \(T^3v = 0\).

(a) Prove that \(\{v,Tv,T^2v\}\) is linearly independent.

Let \(\lambda_1v+\lambda_2Tv+\lambda_3T^2v=0\Rightarrow\lambda_1v+\lambda_2\cdot\left.T\left(v\right.\right)+\lambda_3\cdot T\left(T\left(v\right)\right)=0\)

Since we know \(T(0)=0\), then \(T(\lambda_1v+\lambda_2\cdot\left.T\left(v\right.\right)+\lambda_3\cdot T\left(T\left(v\right)\right))=0\)

Then \(\lambda_1T(v)+\lambda_2T(T(v))+\lambda_3T(T(T(v))=0\), since \(T^3v = 0\).

Then \(\lambda_1T(v)+\lambda_2T(T(v))=0\), then \(T(\lambda_1T(v)+\lambda_2T(T(v)))=0\)

Then \(\lambda_1T(T(v))+\lambda_2T(T(T(v))=0\), since \(T^3v = 0\).

Then \(\lambda_1T(T(v))=0\), since \(T(T(v))\neq 0\), then \(\lambda_1=0\), then \(\lambda_2=0\), then \(\lambda_3=0\)

(b) Prove that there exists a basis \(B\) of \(V\) such that

\[ [T]_B = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 0 & 0 \end{pmatrix}. \]

Since \(\{v,Tv,T^2v\}\) is linearly independent and have dimension \(3\), then it is a basis.

\(T(v)=0\cdot v+1\cdot Tv+0\cdot T^2v\)

\(T\left(T\left(v\right)\right)=0\cdot v+0\cdot Tv+1\cdot T^2v\)

\(T\left(T^2\left(v\right)\right)=0\cdot v+0\cdot Tv+0\cdot T^2v\)

Thus there exists

(c) Find a basis and dimension for \(\ker(T)\) and \(\mathrm{Im}(T)\).

Since \(T^2v\in\ker(T)\) and \(Tv,T\left(Tv\right.)\in\mathrm{Im}(T)\)

Then we know \(Tv\neq \lambda T(T(v))\), since problem 1

Then since the dimension is \(3\)

The basis of kernel is \(\{Tv^2\}\) with dimension 1

The basis of image is \(\{Tv,Tv^2\}\) with dimension 2

Follow-up Questions:

  1. Rate the difficulty of the given problem. (1 - very easy, 9 - very hard)
    1 2 3 4 5 6 7 8 9

    What helped you solve the problem?

    The first problem is the hint for the whole problem, also the definition of kernel and image listed

  2. Suppose the problem is given in the exam, and is worth 20 points. How many points do you think you gained for your solution? Please explain.

    18

    The whole solution i think it is right, but need more explanations and more detailed words

  3. Would you agree to participate in a short follow-up interview by Zoom?
    Yes No

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