TD 1: Bases de l`algèbre linéaire

C
R+
x0(t) + pt2+e−t5x(t) = 0.
x0(t) + pt2+e−t5x(t) = 1.
E F G E
F∩G E
F+G E E F ∪G
F∪G E F ⊂G G ⊂F
E F1,...Fnn E F1∪F2. . . ∪Fn
i Fi⊃Sj6=iFj
R2R2
{((0, y)|y∈R}
{((0, y)|y>0}
{((0, y)|y∈R}∩{((x, 0)|x∈R}
{((0, y)|y∈R}∪{((x, 0)|x∈R}
{((x, y)||x|=|y|}
{((x, y)|x=y}
E[0,1] F([0,1],R
E
E F1={f∈E|f(x) = 0,∀x∈[0,1/3]}F2={f∈
E|f(x)=0,∀x∈]1/3,2/3]}F3={f∈E|f(x) = 0,∀x∈]2/3,1]}
E F1, F2, F3
V ect({t7→ cos(kt)|k6n}) = V ect({t7→ cos(t)k|k6n})
C
x0+ 2tx = 0

{(x, y, z)∈R3|z= 3x+ 5y}
n
Rn0 1
2n−1
{(1,0,0),(1,0,1),(0,0,5)} ⊂ R3
{(1,0,0),(1,0,1),(0,2,5)} ⊂ R3
{x7→ ex, x 7→ e2x}⊂F(R,R)
{x7→ cos(x), x 7→ cos2x, x 7→ cos3x, . . . cos(nx)} ⊂ F(R,R)n∈N∗
u=
0
3
1
v=
−1
h
1
3
w=
k
1
h
3
(h, k)∈R2{u, v, w}h= 1 k= 1
n+ 1
Rn[X]
E n F E F
n dim(F) = n⇔E=F
a0,...ann+ 1 b0, . . . , bnn+ 1
n
n+ 1
(L0,...Lni∈[|0, n|]
Li(ai)=1 Li(aj)=0 j6=i
Rn[X]
P∈Rn[X]i P (ai) = bi
R[X]