Negation of converse of a theorem related to linear independence

Question

• Theorem $T:\quad$ Let $S = \{v_1, v_2,..., v_p\}$ be a set of vectors in $\mathbb R^m$. If $p>m$, then this set is linearly dependent.

Theorem $T$ can be proved to be true.

• Converse $T_c$ of theorem $T:\quad$ If the set $S$ is linearly dependent then $p > m$.

The converse of theorem $T$ is not a true statement. Does it mean that its negation is true?

• Negation $T_{nc}$ of $T_c:\quad$ The set $S$ is linearly dependent and $p \le m$.

In simple words, what does $T_{nc}$ mean, and how to prove it?

Answer 1

The converse of theorem $T$ is not a true statement. Does it mean that its negation is true?

Negating a statement logically flips its truth value; this means that even if you abstractly redefine the phrases (e.g., 'linearly dependent') and non-logical symbols (e.g., the 'less than' symbol) within it, the statement and its negation continues to have opposite truth values. So, the negation of a false statement has to be true.

• Theorem $T:\quad$ Let $S = \{v_1, v_2,..., v_p\}$ be a set of vectors in $\mathbb R^m$. If $p>m$, then this set is linearly dependent.
• Converse $T_c$ of theorem $T:\quad$ If $S$ is linearly dependent then $p > m$.

You have not fully stated the converse $T_c:$ the part about $S$ being an arbitrary set of $p$ vectors in $\mathbb R^m$ is important, because in full: $$\forall p{\in}\mathbb Z_0^+\quad\forall m{\in}\mathbb Z^+\\\forall S{\in}\{\text{sets of $p$ vectors in }\mathbb R^m\}\;(S\text{ is linearly dependent}\implies p>m).\tag1$$

• Negation $T_{nc}$ of $T_c:\quad$ $S$ is linearly dependent and $p \le m$.

Consequently, your negation is incomplete; compare it with the full negation $$\exists p{\in}\mathbb Z_0^+\quad\exists m{\in}\mathbb Z^+\\\exists S{\in}\{\text{sets of $p$ vectors in }\mathbb R^m\}\;(S\text{ is linearly dependent and } p\le m).\tag{1n}$$

how to prove $T_{nc}$ ?

As suggested by Anne Bauval, just exhibit $(p,m,S)=\left\{1,1,\{0\}\right\}.$

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Addendum

The statement I quoted in the question is given as such in a textbook, without quantifiers.

In my second paragraph above, the point is that the "Let $S = \{v_1, v_2,..., v_p\}$ be a set of vectors in $\mathbb R^m$" part of theorem $T$ is understood to mean that $S$ is an arbitrary set of $p$ vectors in $\mathbb R^m,$ and that this stipulation is equivalent to the sequence of universal quantifiers that I made explicit in statement $(1).$

so that negation, converse and all are easy?

Another point worth noting is that when taking the converse or contrapositive, the external quantifiers (the quantifiers outside the "if..., then...") are unaltered, like in the above.