(8) Suppose e, ,em) is an orthonormal list of vectors in V, and v E V. Prove that , e2 v, em)2 if...
5.3.20 Suppose that T E (V, W) has an SVD with right singular vectors e1,..., en E V, left singular vectors fı,. . m E W, and singular values ơi > > ơr > 0 (where r = rank T). Show that: (a) ) is an orthonormal basis of range T. (b) (er+1.. em) is an orthonormal basis of ker T (c) (frt.. .fi) is an orthonormal basis of ker T. (d) (e,...,er) is an orthonormal basis of range T....
Problem 6* (Optional). Suppose ej,..., en is an orthonormal basis of V and v, ...,Vn are vectors in V such that lle; - v, 1 < 1 h for each j. Prove that V1, ..., Vn is a basis of V. In other words, if you perturb an orthonormal basis slightly, you still have a basis.
II. Operators on Hilbert Space 3. Suppose {e,,e2, ) is an orthonormal basis for It', and for each n there is a vector Ae, in such that Σ ll Aenkoo. Show that A has an unique extension to a bounded operator on II. Operators on Hilbert Space 3. Suppose {e,,e2, ) is an orthonormal basis for It', and for each n there is a vector Ae, in such that Σ ll Aenkoo. Show that A has an unique extension to...
2. Suppose that V is an inner product space. (i) Prove that, for any vectors 01, 02 € V, || 0111? + || 0,2||2 = || v1 + v2||2 + || 01 – v2||2 2 (ii) Prove that, for any vectors V1, V2 € V, if v, and v, are orthogonal then || 01 || + || 112 || 2 = || 01 + 02||2.
suppose that s=(v1,v2,......vm) is a finite set of linearly independent vectors in V, and w ∈ V some other vector. Let T= S ∪ (W). Prove that T is not linearly independent if and only if w∈ span(s).
Question 6) (9 points) Prove each of the following statements. (a) Suppose that the vectors {v, w, u} are linearly independent vectors in some vector space V. Prove then that the vectors {v + w, w + u,v + u} are also linearly independent in V. (b) Suppose T is a linear transformation, T: P10(R) → M3(R) Prove that T cannot be 1-to-1 (c) Prove that in ANY inner product that if u and w are unit vectors (ie ||vl|...
#8 6.4.8 Question Help 1 The vectors v1 1 -2 and V2 form an The orthonormal basis of the subspace spanned by the vectors is O. (Use a comma to separate vectors as needed.) 5 3 orthogonal basis for W. Find an orthonormal basis for W.
(a) Find an orthonormal basis for the linear subspace V of R4 generated by the vectors 1 1 1 1 2 (b) What is the projection of the vector on the linear subspace V?
(2) Suppose that W is finite dimensional and T E (V, W). Prove that T is injective if and only if there exists SEC(W, V) such that ST is the identity map on V. (2) Suppose that W is finite dimensional and T E (V, W). Prove that T is injective if and only if there exists SEC(W, V) such that ST is the identity map on V.
6. (i) Prove that if V is a vector space over a field F and E is a subfield of F then V is a vector space over E with the scalar multiplication on V restricted to scalars from E. (ii) Denote by N, the set of all positive integers, i.e., N= {1, 2, 3, ...}. Prove that span of vectors N in the vector space S over the field R from problem 4, which we denote by spanr N,...