Problem 3.2. Let {v1,...,vn} be a basis for V over R. 1. Explain why there must

Question

Problem 3.2. Let {v1,...,vn} be a basis for V over R.

1. Explain why there must exist n linear functions {f1,...,fn} such that, for each i =
1,...,n, fi(vi) = 1 and fi(vj) = 0 for j = i.

2. Prove that the linear functions {f1, . . . , fn} defined in the previous part form a
basis for V ? over R. This basis is called the standard dual basis to {v1, . . . , vn}.

Note: There are two things you must prove. First show that the set {f1, . . . , fn} is
linearly independent; second, show that any linear function f ? V ? can be written as
a linear combination of the linear functions f1, . . . , fn. (Hint: To show these
conditions, consider the values of linear functions on the basis {v1, . . . , vn} for V ).

Explanation / Answer

Here is how you construct the required function fi (i will construct f1) Consider the vector v v = a1 * v1 + ... an * vn now f1: V -> R st f1(v) = a1 (this is uniquely defined since vi's are a basis) This fn is also linear. So this function satisfies the condition for the funct. f1 similarly construct fis Now that they are a basis 1. lin independent: Consider any function f = b1*f1 + .. bn*fn Now say this f=0 which means that the value of f is zero for all vectors So f(v1) = b1 * f1(v1) + b2*f2(v1) + .. bn*fn(v1) = b1 So if f(v1) = 0 then b1 =0 Hence if f is zero for all the vectors v then all the bi's are zero hence its lin. independent. 2. consider any linear fn f:V->R That means that f(a+b) = f(a)+f(b) and f(ca) = cf(a) where c is a scalar say v = a1 * v1 + ... an * vn So f(v) = f(a1 * v1 + ... an * vn) = a1*f(v1) + .. an*f(vn) f1(v) = a1, ... hence f(v) = f(v1)*f1(v) + .. f(vn)*fn(v) here f(vi) are all scalars Hence f is a linear combination of fis Hence done if you have any doubts message me

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