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6. (10) Show that if W is a k-dimensional subspace of an inner product space V...
4. Let TV - V be a linear operator on a finite dimensional inner product space V and P be the orthogonal projection of V onto the subspace W of V. a) Show that is invariant under T if and only if PTP = TP. b) Show that w and we are both invariant under 7 If and only if PT = TP
Suppose V is a finite dimensional inner product space, and dim V = n. If is an orthogonal subset of V, prove that a. W can be extended to an orthogonal basis for V. b. is an orthogonal basis for c.
Let V be a finite-dimensional vector space, and let B be a basis of V. Show that there is an inner product on V for which B is orthonormal.
(7) Let V be a finite-dimensional vector space over F, and PE C(V) In this question, we will show that P is an orthogonal projection if and only if P2P and PP It may be helpful to recal that P is the orthogonal projection onto a subspace U if and only if (1) P is a projection, and (2) ran(P)-U and null(P)U (a) Prove that if P is an orthogonal projection, then P2P and P is self-adjoint Hint: To show...
1). Let V be an n-dimensional inner product space, let L be a linear transformation L : V + V. a) Define for inner product space V the phrase "L:V - V" is an orthogonal transforma- tion". b) Define "orthogonal matrix" b) If v1, ..., Vn is an orthonormal basis for V define the matrix of L relative to this basis and prove that it is an orthogonal matrix A.
Orthogonal projections. In class we showed that if V is a finite-dimensional inner product space and U-V s a subspace, then U㊥ U↓-V, (U 1-U, and Pb is well-defined Inspecting the proofs, convince yourself that all that was needed was for U to be finite- dimensional. (In fact, your book does it this way). Then answer the following questions (a) Let V be an inner product space. Prove that for any u V. if u 0, we have proj, Pspan(v)...
QUESTION 8 Let (V,<,>) be an inner product space, and P: V – V a linear map. Choose the correct statement(s). Multiple choices might be correct and wrong choices have negative points. if P(V) = < W, V > Wand ||w|= 1, then P is an orthogonal projection. if P is an orthogonal projection, then < V- P(V), W> = 0 for any VEV, welmP. fW= Im P and {W 1,...,Wx} is an orthonormal basis for W then P(V) =...
Could someone give me the definitions for these ? You don't need to go into details. just a brief def would do. and pls answer ALL. Thank you Definitions for The abstract definitions of 0 and -in a vector space. - Kernel and image of a linear transformation Span, linear independence, subspace, basis, dimension, rank in the context of an abstract vector space Coordinates of a "vector" with respect to a basis Matrix of a linear transformation with respect to...
3. Let V be a finite dimensional inner product space, and suppose that T is a linear operator on this space. (i) Let B be an ordered orthonormal basis for V and let U be the linear operator on V determined by [U19 = (T);. Then, for all 01,09 € V, (01, T(02)) = (U(V1), v2) (ii) Prove that the conclusion of the previous part does not hold, in general, if the basis 8 is not orthonormal.
(1 point) Are the following statements true or false? ? 1. The best approximation to y by elements of a subspace W is given by the vector y - projw(y). ? 2. If W is a subspace of R" and if V is in both W and Wt, then v must be the zero vector. ? 3. If y = Z1 + Z2 , where z is in a subspace W and Z2 is in W+, then Z, must be...