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Mr Davis 97

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In summary, the conversation discusses the difference between linear independence of two sets of functions, ##\{x,e^x\}## and ##\{ex,e^x\}##. Both sets only have the trivial solution when represented as a linear combination equal to zero. However, the definition of linear independence states that for all x in an interval I, the only solution should be the trivial one. The first set meets this condition since x and e^x never intersect, but the second set does not, as e(1) is a multiple of e^1 at x=1. However, this does not violate the definition since the solution only works for one particular value of

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Mr Davis 97

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Stephen Tashi

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Intersection or non-intersection is irrelevant. The graphs of ##f_1(x) = x^2+1 ## and ##f_2(x) = 2x^2+2## don't intersect, but ## (-2) f_1(x) + (1)f_2(x) = 0 ## at each value of ##x##.Mr Davis 97 said:In the first case, this is obvious since x and e^x never intersect, and so cannot be multiples of each other.

No. At x = 1, we have ##c_1 (ex) + c_2(e^x) = 0## for ##c_1 = 1## and ##c_2= -1##, but those values ##c_1, c_2## are not solutions that apply to each value of ##x## in some interval. They only work at one particular value of ##x##.However, doesn't the latter case violate this definition since ##e(1)## is a multiple of ##e^1##?

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mfb

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I don't understand that argument. In fact, if two functions intersect (but are not identical), they cannot be multiples of each other. If they never intersect, they can (don't have to) be multiples of each other.Mr Davis 97 said:In the first case, this is obvious since x and e^x never intersect, and so cannot be multiples of each other.

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lurflurf

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(-1)e x+(1)e^x=0 if x=1 but not for all x

that is confusing because the "for all x" applies to

##c_1f_1 + c_2f_2 + ... + c_nf_n = 0##

not

there exists only the trivial solution

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Ssnow

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Mr Davis 97 said:"Two functions are linearly independent on the interval II if there exists only the trivial solution to c1f1+c2f2+...+cnfn=0c_1f_1 + c_2f_2 + ... + c_nf_n = 0 for all x in II.

you said correctly at the end '' for every ##x \in I## '' this must happen uniformly in ##I## ...

Linear independence of functions refers to a set of functions that cannot be written as a linear combination of each other. This means that no function in the set can be created by multiplying any of the other functions by a constant and adding them together.

To determine if a set of functions is linearly independent, you can use the Wronskian test. This involves calculating the determinant of a matrix formed by the derivatives of the functions. If the determinant is non-zero, the functions are linearly independent. If the determinant is zero, the functions are linearly dependent.

Yes, it is possible for a set of functions to be linearly independent on one interval and linearly dependent on another. This is because the coefficients used to create a linear combination of the functions can change depending on the interval.

Linear independence is an important concept in mathematics because it allows us to solve systems of linear equations and find unique solutions. It is also used in fields such as differential equations, linear algebra, and functional analysis.

No, two functions cannot be linearly independent if they have the same shape but different amplitudes. This is because one function can be created by scaling the other, making them linearly dependent. Linear independence requires that the functions are truly distinct and cannot be created by scaling or combining each other.

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