ORTHOGONALITY AND LEAST SQUARES

Orthogonal Vectors Norm Unit Vector Orthonormal Vectors Orthogonal Complement Orthogonal Projection Cauchy-Schwarz Inequality Angle Between Two Vectors Gram-Schmidt Process QR Factorialization Transpose Symmetric and Skew-symmetric Matrices Orthogonal Matrices Least Squares Solutions Matrix of an Orthogonal Projection

Orthonormal Vectors How to Examples Exercise How to: Orthonormal vectors are a set of vectors, v1, v2, ... , vM, such that:      1) All vectors are unit vectors, norm(vI) = 1 and      2) All vectors are orthogonal to one another, dot(vI, vJ) = 0 (I does not equal J) An orthonormal basis is simply a basis composed of orthonormal vectors      Examples: Let us show that the vectors v1 = [0.6; 0; -0.8], v2 = [-0.8; 0; -0.6], and v3 = [0; 1; 0] are orthonormal to each other. First, we need to show that they are all unit vectors:      norm(v1) = 1      norm(v2) = 1      norm(v3) = 1 Now, we need to show that they are all orthogonal to one another:      dot(v1, v2) = 0      dot(v1, v3) = 0      dot(v2, v3) = 0 Thus, we have shown that v1, v2, and v3 are orthonormal vectors in R3, and in fact they form an orthonormal basis for R3. Exercise: