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Suppose a linear transformation T : $V(R^3)$ $\,\to\,$ $W(R^3)$ and in this transformation no vector is mapped to zero so the null space has only zero element.

Now suppose that elements in the range are linearly dependent and out of 3 only 2 are linearly independent, then range of T will be 2, but from formula range(T) + nullity(T) = dimension(V),

so from here r(T) + n(T) = 2, and dim (V) =3, which fails this relation. where I am wrong?

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1 Answer 1

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If $T: \mathbb R^3 \to \mathbb R^3$ is linear and $n(T)=0,$ then $r(T)= \dim (V)=3.$

Conclusion: $r(T)=2$ is not possible !

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  • $\begingroup$ please tell me r(T) depends on whether the elements in the range are linearly independent? $\endgroup$
    – sawan kt
    Sep 29, 2020 at 5:45
  • $\begingroup$ Since $r(T)=3$ we have that the range of $T$ is $=\mathbb R^3,$ $\endgroup$
    – Fred
    Sep 29, 2020 at 5:47

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