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-mat

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) 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}) \dots

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 \dots

\dots

\dots

\dots

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 \dots

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 \dots

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})