$$ f(x)=x^{2}-1 \quad g(x)=-k x^{2}+1 \quad h(x)=k x^{2}+k $$ di $$ \mathbb{R}_{\leqslant 2}[x] $$ e sia $$ B $$ il sottospazio da essi generato. (i) Per quali valori di $$ k $$ i polinomi $$ f(x), g(x) $$ e $$ h(x) $$ sono una base di $$ B $$ ? (ix Calcolare la dimensione di $$ B $$ al variare di $$ k \in \mathbb{R} $$. (idi) Per quali valori di $$ k $$ il polinomio $$ a(x)=2-k x+k x^{2} $$ appartiene a $$ B $$ ?

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Park Whittaker · Accepted Answer

For $$ k 
eq 0 $$, the polynomials $$ f(x) = x^2 - 1 $$, $$ g(x) = -kx^2 + 1 $$, and $$ h(x) = kx^2 + k $$ are linearly independent and form a basis for $$ B $$. The dimension of $$ B $$ is 3 for $$ k 
eq 0 $$ and 1 when $$ k = 0 $$. The polynomial $$ a(x) = 2 - kx + kx^2 $$ belongs to $$ B $$ only when $$ k = 0 $$.

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