E is a vectorial plan in R with a base B=(i^→ ,j^→ ). f is an endomorphism of E defined ∀ u^→ =xi^→ +yj^→ by f(u^→ )=(−7x−12y)i^→ +(4x+7y)j^→ . 1) Determinate f(i^→ ) and f(j^→ ) then write the matrice of f in (i^→ ,j^→ )base.


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Question Number 86021 by mathocean1 last updated on 26/Mar/20

$E i s a v e c t o r i a l p l a n i n R w i t h a b a s e$ $B = (\vec{i}, \vec{j}) . f i s a n e n d o m o r p h i s m o f E$ $d e f i n e d \forall \vec{u} = x \vec{i} + y \vec{j} b y f (\vec{u}) = (- 7 x - 12 y) \vec{i} + (4 x + 7 y) \vec{j} .$ $1) D e t e r m i n a t e f (\vec{i}) a n d f (\vec{j}) t h e n$ $w r i t e t h e m a t r i c e o f f i n (\vec{i}, \vec{j}) b a s e .$
Commented by mathocean1 last updated on 27/Mar/20
$2) The following question is: Determinate the matrice of g=fof i found (((1 0)),((0 1)) ) 3)show that g(i^→ )=i^(→ ) and g(j^→ )=j^→ i showed it. 4) calculate fof(u^→ )... Can you help me for this please... ... ... ... 5) E_1 ={u^→ ∈ E/ f(u^→ )=u^→ } E_2 ={u^(→ ) ∈ E/f(u^→ )=−u^→ } •Show that E_1 and E_2 are vectorial lines. •then give one of theirs bases e_(1 ) ^→ and e_2 ^→ . •show that (e_1 ^→ ,e_2 ^→ ) is a base of E. ∗Determinate the matrice of f in this base. Please i need your help sirs...$
$2) T h e f o l l o w i n g q u e s t i o n i s :$ $D e t e r m i n a t e t h e m a t r i c e o f g = f o f$ $i f o u n d (\begin{matrix} 1 0 \\ 0 1 \end{matrix})$ $3) s h o w t h a t g (\vec{i}) = \vec{i} a n d g (\vec{j}) = \vec{j}$ $i s h o w e d i t .$ $4) c a l c u l a t e f o f (\vec{u}) . . .$ $C a n y o u h e l p m e f o r t h i s p l e a s e . . .$ $. . .$ $. . .$ $. . .$ $5) E_{1} = {\vec{u} \in E / f (\vec{u}) = \vec{u}}$ $E_{2} = {\vec{u} \in E / f (\vec{u}) = - \vec{u}}$ $∙ S h o w t h a t E_{1} a n d E_{2} a r e v e c t o r i a l$ $l i n e s .$ $∙ t h e n g i v e o n e o f t h e i r s b a s e s {\vec{e}}_{1} a n d {\vec{e}}_{2} .$ $∙ s h o w t h a t ({\vec{e}}_{1}, {\vec{e}}_{2}) i s a b a s e o f E .$ $* D e t e r m i n a t e t h e m a t r i c e o f f i n t h i s$ $b a s e .$ $P l e a s e i n e e d y o u r h e l p s i r s . . .$
Commented by mathocean1 last updated on 26/Mar/20

$c a n y o u h e l p m e p l e a s e . . .$
Commented by abdomathmax last updated on 26/Mar/20

$i (1, 0) \Rightarrow f (i) = - 7 i + 4 j \Rightarrow f (i) (\begin{matrix} - 7 \\ 4 \end{matrix})$ $j (0, 1) \Rightarrow f (j) = - 12 i + 7 j \Rightarrow f (j) (\begin{matrix} - 12 \\ 7 \end{matrix})$ $M_{f} (i, j) = (\begin{matrix} - 7 - 12 \\ 4 7 \end{matrix})$
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