Assigment-2 1. Let U = (, , 4) and 1 = (1, 1, 13). Show that...

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Assigment-2 1. Let U = (₁, ₂, 4₂) and 1 = (1₁, 1₂, 13). Show that the expression does not define an inner product on R³, and list all inner product ariom that fail to hold @ <0,0) = 4₁ ²0₁² +4²1₂²2² +4²13² 1 (u, v) = 4₁00₁ - 4₂ 1₂ +13 43 Из Из 2. @Prove that it is a fixed vector in a seal innes product Space V, then the mapping by T (₂) = (1, 0) is a T: VR defined linear transformation 3 Let V = R²³ have the Euclidean inner product, and let v = (1, 0, 2). Compute T(ILD)) Let V=P have the evalution inner Product at the points x6 =1, 7₁=0, 2₂ = -1 and let v = 1+x. Compute T(x+2²) 3. Do there exist scalars k and I such that the vectors P₁ = 2+kx+62², B₂ =l+5x+3x², P₂ = 1+2x+32² are mutually orthogonal with respect to the standard inner product of Pa 1. 4. Find a basis for the oothogonal Complement of the subspace of R² spanned by the vectors V₁ = (1, 4, 5, 2), V₂ = (2,1,3,0), V₁ = (-1,3,2,2) M₁ = (1, 4, 5, 6, 9), V₂ = ( 3₁-2, 114, -1 (a) N3 = (-1,0, -1, -2, -1), V₁ = (2, 3, 5, 7, 8) = 5. Let v be that if u and I are an immer product Space. Show orthogonal unit vectors in V, then 11u-v1l = √₂. 6. to Use the Cauchy - Schwarz inequality Prove that for all real values of a, b, 0, 2 (acro- a Cos 0 + b sino) ≤ a² + b² 7. Let R¹ have the Euclidean, inner product, the the Gram-Schmidt process to transtorm the basis & 1₁, 4₂, 43, 4) into an ortho normal basis U₁ = (0, 2, 1, 0), 4₂ = (1,-1,0,0) Ug = (1, 2, 0,-1) g. Find 8. Obtain the Column vectors of 8 by the Gram Schmidt process to the. applying the Column vector of A. Find a &R decomposition of the matrix A. A= [2 a &R A = ₁ U4 = (1,0,0,1), A - 3, A= 3. -1 decomposition of the matory 1 1 1 1 10. Determine whether the set of vector is Orthogonal and whether it is or the mormed with respect to the Euclidean inner produce in R² @ (01), (20) Ⓒ (-₂-ti), (+₂^ti) (0,0) / (0,1) Assigment-2 1. Let U = (₁, ₂, 4₂) and 1 = (1₁, 1₂, 13). Show that the expression does not define an inner product on R³, and list all inner product ariom that fail to hold @ <0,0) = 4₁ ²0₁² +4²1₂²2² +4²13² 1 (u, v) = 4₁00₁ - 4₂ 1₂ +13 43 Из Из 2. @Prove that it is a fixed vector in a seal innes product Space V, then the mapping by T (₂) = (1, 0) is a T: VR defined linear transformation 3 Let V = R²³ have the Euclidean inner product, and let v = (1, 0, 2). Compute T(ILD)) Let V=P have the evalution inner Product at the points x6 =1, 7₁=0, 2₂ = -1 and let v = 1+x. Compute T(x+2²) 3. Do there exist scalars k and I such that the vectors P₁ = 2+kx+62², B₂ =l+5x+3x², P₂ = 1+2x+32² are mutually orthogonal with respect to the standard inner product of Pa 1. 4. Find a basis for the oothogonal Complement of the subspace of R² spanned by the vectors V₁ = (1, 4, 5, 2), V₂ = (2,1,3,0), V₁ = (-1,3,2,2) M₁ = (1, 4, 5, 6, 9), V₂ = ( 3₁-2, 114, -1 (a) N3 = (-1,0, -1, -2, -1), V₁ = (2, 3, 5, 7, 8) = 5. Let v be that if u and I are an immer product Space. Show orthogonal unit vectors in V, then 11u-v1l = √₂. 6. to Use the Cauchy - Schwarz inequality Prove that for all real values of a, b, 0, 2 (acro- a Cos 0 + b sino) ≤ a² + b² 7. Let R¹ have the Euclidean, inner product, the the Gram-Schmidt process to transtorm the basis & 1₁, 4₂, 43, 4) into an ortho normal basis U₁ = (0, 2, 1, 0), 4₂ = (1,-1,0,0) Ug = (1, 2, 0,-1) g. Find 8. Obtain the Column vectors of 8 by the Gram Schmidt process to the. applying the Column vector of A. Find a &R decomposition of the matrix A. A= [2 a &R A = ₁ U4 = (1,0,0,1), A - 3, A= 3. -1 decomposition of the matory 1 1 1 1 10. Determine whether the set of vector is Orthogonal and whether it is or the mormed with respect to the Euclidean inner produce in R² @ (01), (20) Ⓒ (-₂-ti), (+₂^ti) (0,0) / (0,1)

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