Expressing one function as linear combination of others

[koolrizi]

How would you express one function as Linear combination of other? For example the following sets
{1,x+2,3x-5}
{e^x,e^(2x),xe^x,(7x-2)e^x}

How would i go about solving this?
Should I start with equating one function to the other two in first example?
like x+2=1+(3x-5) ?

Thanks

 

[HallsofIvy]

How would you express one function as Linear combination of other? For example the following sets
{1,x+2,3x-5}
{e^x,e^(2x),xe^x,(7x-2)e^x}

How would i go about solving this?
Should I start with equating one function to the other two in first example?
like x+2=1+(3x-5) ?

Thanks

Well, since x+2= 1+(3x-5) is NOT true, that's pretty obviously NOT a good start!

What is a good start is thinking about the DEFINITION of "linear combination". (Thinking about definitions is, in general, a good way to start a problem!)

A "linear combination" of vectors (or functions) is a sum of multiples of them: av₁+ bv₂+ ...

Here, you are looking for a, b, such that 3x-5= a(1)+ b(x+2). Find a and b so that is true for all x. If x= 0, then -5= a+ 2b. If x= 1, then -2= a+ 3b. Solve those equations for a, b.

Similarly, you want to find numbers a, b, c so that
(7x-2)e^x= ae^x+ be^2x+ cxe^x
for all x. You might start by simplifying those by choosing specific values of x. Since they are to be true for all x, it doesn't really matter which you choose. You can choose 3 values of x so that you get 3 equations for a, b, c.

 

[koolrizi]

Thanks HallsofIvy