Finding Dimensions of Vector Subspace Spanning a Set

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To determine the dimensions of the vector subspace spanned by the functions e^x and e^2x, it is essential to check their linear independence. This can be done by setting up the equation ae^x + be^2x = 0 and analyzing the coefficients. Differentiating the equation can provide additional insights, allowing for manipulation to isolate one of the terms. The goal is to establish whether a non-trivial solution exists for the constants a and b. Ultimately, confirming independence will lead to identifying the basis and determining the dimension of the subspace.
Pearce_09
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The functions e^x and e^2x
I have to find the dimmensions of the vector subspace spanning the set.
I understand how to solve other problems like involving matrices and row reducing, but this function i don't know where to start and how to figure out the dimensions. Any help would be greatly appriciated
thanks
adam
 
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You need to show if it is independent or not to find the basis and consequently the dimension. The first equation is ae^x + be^2x = 0; which you know. To find another, try differentiating the equation and then manipulating these equations to isolate for one of the two terms(ie, e^x or e^2x). Since this is a function, The equation should hold true for all values of t. perhaps try some that will give you an answer for one of the constants a or b.
 

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