Question Number 101645 by mathocean1 last updated on 03/Jul/20

E is a vectorial plane in B=(i^→ ,j^→ ) base. f is an endomorphism of E. f(i^→ )=4i^→ −j^→ and f(j^→ )=2i^→ +j^→ . u^→ =xi^→ +yj^→ ∈ E and x,y ∈ R. 1) Determinate f^( −1) (u).

Answered by mathmax by abdo last updated on 03/Jul/20

M(f,B) = (((4 2)),((−1 1)) ) we have u ((x),(y) ) f^(−1) (u)→M^(−1) .u let find M^(−1) M is root of (4−u)(1−u)+2 =0 ⇒ (u−4)(u−1)+2 =0 ⇒u^2 −5u +6 =0 ⇒M^2 −5M +6I =0 ⇒ M^2 −5M =−6I ⇒−(1/6)M.(M−5I) =I ⇒M^(−1) =−(1/6)(M−5I) = (((−(2/3) −(1/3))),(((1/6) −(1/6))) ) + ((((5/6) 0)),((0 (5/6))) ) = (((−(1/6) −(1/3))),(( (1/6) (2/3))) ) let f^(−1) (u) = ((x^′ ),(y^′ ) ) ⇒ ((x^′ ),(y^′ ) ) =M^(−1) .u = (((−(1/6) −(1/3))),(((1/6) (2/3))) ) . ((x),(y) ) = (((−(x/6)−(y/3))),(((x/6) +((2y)/3))) )