Gram Schmidt Process is an algorithm for orthonormalizing vectors in an inner product space. Let’s explain what this algorithm do.
Perpendicularity Relationship between lines in Space
In geometry, one of the primary relationships between two lines in space is perpendicularity. We say that one line is being perpendicular to another line if the first line meets the second line at right angles. Perpendicularity is a symmetric relationship – In other words, if one line is being perpendicular to the second line, the second line is also being perpendicular to the first line. Therefore, You can say that two lines are being perpendicular to each other or the two lines are being perpendicular.
Orthogonality Relationship between Vectors in Inner Product Space
In linear algebra, one of the primary relationships between two vectors is orthogonality. You can think orthogonality relationship between two vectors in inner product space as a generalization of the perpendicularity relationship between two lines in geometry space.
We say that one vector A is orthogonal to another vector B in inner product space if the inner product <A,B>=0. Since inner product is symmetric (<A,B>=<B,A>), if one vector is orthogonal to the second vector, the second vector is also orthogonal to the first vector. Therefore, You can say that two vectors are orthogonal to each other or the two vectors are orthogonal.
orthonormalizing vectors in an inner product space
We say that set of vectors is orthogonal if each pair of vectors in the set are orthogonal.
When we orthonormalizing vectors in an inner product space, we takes a set of vectors that should linearly independent and finite and generate a new set of vectors that is orthogonal and span the original set.
In other words, The purpose of the Gram Schmidt Process is to generate a new set of vectors that is orthogonal and span the input set of the algorithm.
Visualization of Gram Schmidt Process in Plain
In the following movie, we can see visualization of Gram Schmidt Process in plain. In the movie, we can see how we generate a orthogonal set of 2 vectors which span the input set of the given 2 vectors.
Visualization of Gram Schmidt Process in space
In the following movie, we can see visualization of Gram Schmidt Process in space. In the movie, we can see how we generate a orthogonal set of 3 vectors which span the input set of the given 3 vectors.
