Let A be a 3 x 3 orthogonal matrix with column vectors C, C2, C3. Use...

  

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Let A be a 3 x 3 orthogonal matrix with column vectors C₁, C2, C3. Use properties of orthogonal matrices and the inner product to prove that the vector v = c₁ +2c2 + 2c3 has length ||v|| = 3. Provide references to theorems from the course notes for any such properties that you use in your proof. 15 Inner product and orthogonality If u = (u₁,..., un) and v = (v₁,..., Un) are vectors in R" then we define their inner product by Definition 15.1. u v = U₁v₁ + ... + Un Vn Notice that if we write u, v as 1 x n row matrices then u v = uv and if we write them as n x 1 column matrices then u. v = utv. Some important properties of the inner-product are Theorem 15.1. Let u, v, w ER" and a ER, then (a) u. v = v. u, (b) (u + v) wu.w+v.w, (c) (au) v a(u. v), (d) v v≥0, and v v=0 if and only if v = 0. Proof. The first three are straightforward to prove using the definition or the fact that u v = utv and known results from matrix multiplication. The third is also not hard but important so let us check it . V. V = V₁V₁ + V₂V₂ + + Un vn = (v₁)² + (v₂)² + ... + (vn) ² Being a sum of non-negative numbers this is equal to zero if and only if v₁ = V₂ = = Vn = 0. That is if and only if v = 0. In R² and R³ the inner product is the dot product and the quantity √v.v is the length of the vector v. So we make that definition in any dimension. Definition 15.2. If ve Rn we define the length of v to be ||v|| v, u € Rn we define the distance between v and u to be ||vu||. 0 = arccos = Notice that a vector has length zero if and only if it is the zero vector. Recall that in R2 if we have two vectors v and u then v u = ||v||||u|| cos(0) where is the angle between them as in Figure 15.1. In Rn we don't really know what the angle between two vectors is but we do know v u, |v|| and ||u||. So we can try and use these to define the angle. Obviously we would like to define v.u ||v||||u|| v. v. If Let A be a 3 x 3 orthogonal matrix with column vectors C₁, C2, C3. Use properties of orthogonal matrices and the inner product to prove that the vector v = c₁ +2c2 + 2c3 has length ||v|| = 3. Provide references to theorems from the course notes for any such properties that you use in your proof. 15 Inner product and orthogonality If u = (u₁,..., un) and v = (v₁,..., Un) are vectors in R" then we define their inner product by Definition 15.1. u v = U₁v₁ + ... + Un Vn Notice that if we write u, v as 1 x n row matrices then u v = uv and if we write them as n x 1 column matrices then u. v = utv. Some important properties of the inner-product are Theorem 15.1. Let u, v, w ER" and a ER, then (a) u. v = v. u, (b) (u + v) wu.w+v.w, (c) (au) v a(u. v), (d) v v≥0, and v v=0 if and only if v = 0. Proof. The first three are straightforward to prove using the definition or the fact that u v = utv and known results from matrix multiplication. The third is also not hard but important so let us check it . V. V = V₁V₁ + V₂V₂ + + Un vn = (v₁)² + (v₂)² + ... + (vn) ² Being a sum of non-negative numbers this is equal to zero if and only if v₁ = V₂ = = Vn = 0. That is if and only if v = 0. In R² and R³ the inner product is the dot product and the quantity √v.v is the length of the vector v. So we make that definition in any dimension. Definition 15.2. If ve Rn we define the length of v to be ||v|| v, u € Rn we define the distance between v and u to be ||vu||. 0 = arccos = Notice that a vector has length zero if and only if it is the zero vector. Recall that in R2 if we have two vectors v and u then v u = ||v||||u|| cos(0) where is the angle between them as in Figure 15.1. In Rn we don't really know what the angle between two vectors is but we do know v u, |v|| and ||u||. So we can try and use these to define the angle. Obviously we would like to define v.u ||v||||u|| v. v. If