4 equivalently range T = ker T)4) Prove that there does not exist a linear map T: RS → R$ such that range T = null T (orequivalently range T = ker T).5) Suppose DE L(R[x], R2[x]) is the differentiation map defined by Dp = p'. RecallF,[x] is the vector space of all polynomials over a field F of degree at most n and has au) Find a haaiam Lul

4 We have to prove that there does not exist a linear map T:R5→R5such that rangeT=nullT. We know…

Answered: 4) Prove that there does not exist a…

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equivalently range T = ker T)
4) Prove that there does not exist a linear map T: RS → R$ such that range T = null T (or
equivalently range T = ker T).
5) Suppose DE L(R[x], R2[x]) is the differentiation map defined by Dp = p'. Recall
F,[x] is the vector space of all polynomials over a field F of degree at most n and has a
u) Find a haaia
m Lul

Expert Solution

Step 1

$(4)$ We have to prove that there does not exist a linear map $T : R^{5} \to R^{5}$ such that $r a n g e T = n u l l T$ .

We know that the range-nullity theorem is given by

$R a n k (T) + n u l l i t y (T) = \dim (V)$ , where $V$ is the vector space.

Here, $V = R^{5}$

$\dim (V) = 5$

Let us suppose $r a n g e T = n u l l T = m$

$\begin{array}{l} \Rightarrow m + m = 5 \\ \Rightarrow 2 m = 5 \\ \Rightarrow m = \frac{5}{2} \\ \Rightarrow m = 2.5 \end{array}$

Which is not possible.

Hence, it contradict the statement $r a n g e T = n u l l T = m$ .

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4) Prove that there does not exist a linear map T: R - R such that range T null T (or equivalently range T = ker T'). %3D