Linear Algebra Question Inner Products Use the inner product

Linear Algebra Question; Inner Products

Use the inner product on R² generated by the matrix

w is a vector

Solution

We know that if A is an invertible n x n matrix and u, v are vectors in Rⁿ, then the inner product on Rⁿ generated by A is defined as <u, v> = Au · Av Here u=v = w so that <w.w> = Aw. Aw. Hence ||w|| = (<w.w>)^1/2 = (Aw. Aw)^1/2

Here Aw = (41,50)^T so that Aw. Aw = (41,50)^T. (41,50)^T = 41*41+50*50 = 1681+2500 = 4181.Therefore, ||w|| = 4181