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

Orthogonal Vectors How to Examples Exercise How to: We actually already reviewed the concept of orthogonal vectors when we spoke of perpendicular vectors in the Dot Product section. Orthogonal is simply another way of saying perpendicular. It's hard to imagine what two perpendicular vectors would look like in a dimension higher than 3. Recall that two vectors are perpendicular (orthogonal) if their dot product is equal to zero:      dot(vector1, vector2) = 0      Examples: Let's work with vectors v1 = [1; 2; 3], v2 = [-1; 2; -1], and v3 = [-1; 1; 3]. Computing dot products we find that:      dot(v1, v2) = 0      dot(v2, v3) = 0      dot(v1, v3) = 10 So v1 is orthogonal to v2, v2 is orthogonal to v3, but v1 is NOT orthogonal to v3. This shows us that the property of orthogonality is not transitive. Exercise: