Family vektorspace- linearly independent

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[h=1] [/h][h=1] [FONT=&quot]Hi, can u help me, i have a lot of tasks in Algebra and i don`t understand, how can i prove them. I understand the tasks but proof is yet impossible for now. Can u show me, how i must proove these type of tasks. So the task:

For every quantity M "different to 0" ist the quantity depiction(M;R) of all depictions f:M"arrow"R a R-vektorspace via:
(f + g)(x) := f(x) + g(x); (af)(x) := af(x) (x [/FONT][FONT=&quot]“is from”[/FONT][FONT=&quot] M)
(f;g "is from" depiction(M;R), a "is from " R) - that should be prove. U must look and pay attention to t "is from" R und :
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[FONT=&quot]f_[/FONT][FONT=&quot]t[/FONT][FONT=&quot](index t)[/FONT][FONT=&quot](x) :=[/FONT]	[FONT=&quot] 0[/FONT]	[FONT=&quot]x[/FONT]	[FONT=&quot]< t;[/FONT]	[FONT=&quot] [/FONT][FONT=&quot](x [/FONT][FONT=&quot]“is from”[/FONT][FONT=&quot] R) [/FONT]
[FONT=&quot] 1 [/FONT]	[FONT=&quot]x[/FONT]	[FONT=&quot] =>[/FONT]	[FONT=&quot]t[/FONT]

u should define element f_t "is from" depiction(R;R). U schould solve - Is this Family (f_t)"(small t)" " is from" R linearly independent in R-Vektorspace depiction (R;R)?
Aufgabe 18[/FONT]

[/h][h=1][FONT=&quot]Fur• jede Menge M [/FONT][FONT=&quot]“unterschiedlich zu“ 0[/FONT][FONT=&quot] ist die Menge Abb(M; R) aller Abbildungen f : M [/FONT][FONT=&quot]pfleile[/FONT][FONT=&quot] R ein R-Vektorraum via[/FONT]

[FONT=&quot](f + g)(x) := f(x) + g(x); (af)(x) := af(x) (x [/FONT][FONT=&quot]“ist von”[/FONT][FONT=&quot] M)[/FONT]

[FONT=&quot](f; g [/FONT][FONT=&quot]“ist von”[/FONT][FONT=&quot] Abb(M; R), a [/FONT][FONT=&quot]“ist von”[/FONT][FONT=&quot] R), das braucht nicht bewiesen zu werden. Betrachte fur• t "ist von" R das durch[/FONT]

[FONT=&quot]f_[/FONT][FONT=&quot]t[/FONT][FONT=&quot](index t)[/FONT][FONT=&quot](x) :=[/FONT]	[FONT=&quot]0[/FONT]	[FONT=&quot]x[/FONT]	[FONT=&quot]< t;[/FONT]	[FONT=&quot] [/FONT][FONT=&quot](x [/FONT][FONT=&quot]“ist von”[/FONT][FONT=&quot] R)[/FONT]
[FONT=&quot](1[/FONT]	[FONT=&quot] x[/FONT]	[FONT=&quot] [/FONT]	[FONT=&quot]t[/FONT]
[FONT=&quot] [/FONT]	[FONT=&quot] [/FONT]	[FONT=&quot] [/FONT]	[FONT=&quot] [/FONT]	[FONT=&quot] [/FONT]

[FONT=&quot]de[/FONT][FONT=&quot]fi[/FONT][FONT=&quot]nierte Element f[/FONT][FONT=&quot]_[/FONT][FONT=&quot]t [/FONT][FONT=&quot]“ist von”[/FONT][FONT=&quot] Abb(R; R). Entscheide, ob die Familie (f[/FONT][FONT=&quot]_[/FONT][FONT=&quot]t)t[/FONT][FONT=&quot] „ist von“[/FONT][FONT=&quot]R linear unabhangig• im R-Vektorraum Abb(R; R) ist.[/FONT]
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