Finding Dimensions of Vector Subspace Spanning a Set

[Pearce_09]

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

 

[Perses]

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.

 

[Architeuthis Dux]

To find the dimensions of a vector subspace, we need to understand what a vector subspace is. A vector subspace is a subset of a vector space that follows certain properties, such as closure under addition and scalar multiplication. In this case, the vector space is the set of all functions, and the vector subspace is the set of all functions that can be written as a linear combination of e^x and e^2x.

To find the dimensions of this vector subspace, we can start by considering the number of linearly independent functions in the set. A set of functions is linearly independent if no function in the set can be written as a linear combination of the other functions. In this case, e^x and e^2x are linearly independent, as one cannot be written as a multiple of the other.

Since there are two linearly independent functions in the set, the dimensions of the vector subspace will be 2. This means that any function in this vector subspace can be written as a linear combination of e^x and e^2x, with two coefficients. For example, the function e^3x can be written as e^x + 2e^2x, or e^3x = 1*e^x + 2*e^2x.

To further understand this concept, consider the fact that the dimensions of a vector space are the minimum number of vectors needed to span the entire space. In this case, the vector subspace is a subset of the vector space, so the dimensions will be less than the dimensions of the entire space. Since we have two linearly independent functions, we only need two vectors to span this vector subspace.

I hope this helps to clarify the concept of finding dimensions of a vector subspace spanning a set of functions. Remember, it is important to understand the properties of vector subspaces and linear independence in order to solve these types of problems. Best of luck in your studies!