

Orthogonal Vectors Norm Unit Vector Orthonormal Vectors Orthogonal Complement Orthogonal Projection CauchySchwarz Inequality Angle Between Two Vectors GramSchmidt Process QR Factorialization Transpose Symmetric and Skewsymmetric Matrices Orthogonal Matrices Least Squares Solutions Matrix of an Orthogonal Projection 
GramSchmidt Process
How to Examples Exercise How to: The GramSchmidt process takes a regular basis of a subspace V and constructs with it an orthonormal basis. So, we are given a regular basis v1, v2, ... , vM, and we want to find w1, w2, ... , wM, an orthonormal basis. Finding w1 is simple (unlike the others), it is simply v1 as a unit vector: w1 = v1/norm(v1) Finding the other vectors is not as simple and requires an additional computation. To find wJ, we have to find the orthogonal projection of vJ onto a subspace of V which is the span of all the vectors before vJ (that is v1, v2, ... vJminus1). We'll call this subspace VJminus1 (since we can't use the '' symbol in the name of variables). So, the orthogonal projection, projVJminus1vJ of vJ onto VJminus1 is given by: projVJminus1vJ = dot(w1, vJ)*w1 + dot(w2, vJ)*w2 + ... + dot(wJminus1, vJ)*wJminus1 Now, we can use this to calculate wJ: wJ = (vJ  projVJminus1vJ)/norm(vJ  projVJminus1vJ) This process can be very exhausting...luckily MATLAB simplifies it for us with QRFactorialization. Examples: Exercise: 