i have a question about proofing but i dont understand exactly, the question asked me whether the following expression is true or not, if it is true i need to prove that but if it is false, give counterexample is enough,
The expression : "If a linear combination of a and b equals 1, then so does linear combination of a^2 and b^2."
Could anyone help me ?
The question is very poor because it is ambiguous. Let $0 \ne a,b \in \Bbb R$ be given, and assume that there exist $\alpha. \beta \in \Bbb R$ such that $\alpha a + \beta b = 1$.
If you are asking whether $\alpha a^2 + \beta b^2 = 1$, then this is clearly false: just take $a = b = \frac 1 2$ and $\alpha = \beta = 1$; then $\alpha a^2 + \beta b^2 = \frac 1 4 \ne 1$.
If you are asking whether there exist $\alpha ', \beta ' \in \Bbb R$ such that $\alpha' a^2 + \beta' b^2 = 1$, then this is true: just choose $\alpha' = \frac \alpha a$ and $\beta' = \frac \beta b$. Then $\alpha' a^2 + \beta' b^2 = \frac \alpha a a^2 + \frac \beta b b^2 = \alpha a + \beta b = 1$.
If $a = 0$ and $b \ne 0$, assume that there exist $\beta$ with $\beta b = 1$. If $b \ne 1$ then clearly it is not true that $\beta b^2 = 1$. Nevertheless, you may find $\beta '$ such that $\beta' b^2 = 1$ - namely, choose $\beta' = \frac 1 {b^2}$.
If $a=b=0$, then the question is trivial.