Consider the following function in R2 v w V1W1 5V2W2 where

Consider the following function in R2: (v, w) = V1W1 + 5V2W2, where v, w R2. Verify that the function in equation (1) defines an inner product. Consider a set S\' = {v1, V2}, where vi = (2,1) and V2 = (5, -2). Show that vi and V2 are orthogonal to each other with respect to the inner product given in equation (1). Write the vector w = (-2,5) as a linear combination of the basis vectors in S.

Solution

c) Let the basis vectors in S be v1 and v2 which span the entire space

<a,b> = a1b1 + 5a2b2

Then in order to write it as basic vectors it must satisfy that

a1b1 = -2

a2b2 = 1

Hence the basis vector woll be (-2 1) and (1 1)

Hence w = <a,b> where a is the vector equal to (-2,1) and b is the vector equal to (1,1)

will they constitute the basis

They are linearly independent, sInce Determinant can\'t be zero and they can generate any vectors hence these vector are the basis for S