![Orthogonality property
Let P and Q be two vectors in Rn , then the differences
of Q and its vector projection onto P is
orthogonal to P.
ie P is orthogonal to [Q-(
𝑄⋅𝑃
𝑃
𝑝
𝑃
]](https://image.slidesharecdn.com/orthoonality-211227041056/85/Orthogonality-6-320.jpg)
![Grahm Schmidt orthogonalization
process
Let P and Q be two non zero vectors then the above orthogonality property
can be used to construct a new vector.
R=[Q-(
𝑄⋅𝑃
𝑃
𝑝
𝑃
]
Which is orthogonal to P
This process of constructing orthogonal vector is known as Grahm Schmidt
orthogonalization process.](https://image.slidesharecdn.com/orthoonality-211227041056/85/Orthogonality-7-320.jpg)
Orthogonality
- 1. Definition Orthogonal Vectors Two vectors u, v are said to be orthogonal provided their dot product is zero: u·v = 0. If both vectors are nonzero (not required in the definition), then the angle θ between the two vectors is determined by cos θ = 𝑢⋅𝑣 𝑢 𝑣 = 0, which implies θ = 90◦. In short, orthogonal vectors form a right angle.
- 2. Unitization/Normalization Any nonzero vector u can be multiplied by c = 1 𝑢 to make a unit vector v = cu, that is, a vector satisfying ||v|| = 1. This process of changing the length of a vector to 1 by scalar multiplication is called Unitization/ Normalization.
- 3. Orthogonal and Orthonormal vectors A set {f1, f2, ..., fm} of non zero vectors in 𝑅𝑛 was called an orthogonal set if fi·fj = 0 for all i ≠ j. An orthogonal set is said to be orthonormal if each vector fm is a unit vector.ie ||fm||=1
- 4. Independence and Orthogonality Theorem 1 :An orthogonal set of nonzero vectors is linearly independent. Proof: Let c1, . . . , ck be constants such that nonzero orthogonal vectors u1, . . . , uk satisfy the relation c1u1 + · · · + ckuk = 0. Take the dot product of this equation with vector uj to obtain the scalar relation c1u1 · uj + · · · + ckuk · uj = 0. Because all terms on the left are zero, except one, the relation reduces to the simpler equation Cj||uj||2 = 0. This equation implies cj = 0. Therefore, c1 = · · · = ck = 0 and the vectors are proved Linearly independent.
- 5. Projecion Let V,W V and w=0. then a real number ‘c’ is called the component of V along W if V-cw is orthogonal to W ie <V-cw,w>=0 <v,w>-c<w,w>=0 c= <v,w> <w,w> Thus if w is any
- 6. Orthogonality property Let P and Q be two vectors in Rn , then the differences of Q and its vector projection onto P is orthogonal to P. ie P is orthogonal to [Q-( 𝑄⋅𝑃 𝑃 𝑝 𝑃 ]
- 7. Grahm Schmidt orthogonalization process Let P and Q be two non zero vectors then the above orthogonality property can be used to construct a new vector. R=[Q-( 𝑄⋅𝑃 𝑃 𝑝 𝑃 ] Which is orthogonal to P This process of constructing orthogonal vector is known as Grahm Schmidt orthogonalization process.