Let v1,..., VN ERN. Define the N x N matrix K by Kij = V ...

 

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Let v1,..., VN ERN. Define the N x N matrix K by Kij = Vị • v; (i,j = 1,..., N). (a) (1 pt.) Find A such that K = AT A. (b) (4 pts.) Show that if v1,..., VN are linearly independent, then K is positive definite. Hint: if the columns of a square matrix are linearly independent, then the matrix is nonsingular. (c) (1 pt.) Show that if v1,..., VN are not linearly independent, then K is semipos- itive definite, but not positive definite. (d) (2 pts.) Show that if K is positive definite, then v1, ..., VN must be linearly independent. (e) (6 pts.) In general, given vectors v1,..., VN E V, one can define the matrix K by Kij So f(x)g(r)dx. Compute K with vị = fi, v2 = f2, v3 = f3, and show that it is nonsingular. (Vi, V3). Consider fi(x) = 1, f2(x) = x, f3(x) = x², and (f, g) = Let v1,..., VN ERN. Define the N x N matrix K by Kij = Vị • v; (i,j = 1,..., N). (a) (1 pt.) Find A such that K = AT A. (b) (4 pts.) Show that if v1,..., VN are linearly independent, then K is positive definite. Hint: if the columns of a square matrix are linearly independent, then the matrix is nonsingular. (c) (1 pt.) Show that if v1,..., VN are not linearly independent, then K is semipos- itive definite, but not positive definite. (d) (2 pts.) Show that if K is positive definite, then v1, ..., VN must be linearly independent. (e) (6 pts.) In general, given vectors v1,..., VN E V, one can define the matrix K by Kij So f(x)g(r)dx. Compute K with vị = fi, v2 = f2, v3 = f3, and show that it is nonsingular. (Vi, V3). Consider fi(x) = 1, f2(x) = x, f3(x) = x², and (f, g) =

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