Section 6.1 -inner Products and Norms

1. 〈x + z,y〉 = 〈x,y〉+ 〈z,y〉 2. 〈cx,y〉 = c〈x,y〉 3. 〈x,y〉 = 〈y,x〉, where the bar denotes complex conjugation. 4. 〈x,x〉 > 0 if x 6= 0 Note that if z is a complex number, then the statement “z ≥ 0” means that z is real and non-negative. Notice that if F = R, (3) is just 〈x,y〉 = 〈y,x〉. Definition. Let A ∈ Mm×n(F ). We define the conjugate transpose or adjoint of A to be the n ×m matrix A∗ such that (A∗)i,j = Ai,j for all i, j. Theorem 6.1. Let V be an inner product space. Then for x,y, z ∈ V and c ∈ F , 1. 〈x,y + z〉 = 〈x,y〉+ 〈x, z〉 2. 〈x, cy〉 = c〈x,y〉 3. 〈x,0〉 = 〈0,x〉 = 0