Please provide an example where u, v ∈ V (V is an inner product space) s.t. ||u+v||^2=||u||^2+||v||^2 but u and v are not orthogonal.
Please provide an example where u, v ∈ V (V is an inner product space) s.t....
8. Show that if [u, v, w\ is an orthogonal set in an inner product space V, then
4. Consider the vector space V = R3 and the matrix 2 -1 -1 2 -1 -1 0 2 We can define an inner product on V by (v, w) = v'Mw. where vt indicates the transpose. Please note this is NOT the standard dot product. It is a inner product different (a) (5 points) Apply the Gram-Schmidt process to the basis E = {e1,e2, e3} (the standard basis) to find an orthogonal basis B.
4. Consider the vector space...
Let V be a real inner product space. Under what condition on u, v E V is the following equality valid: where l 1x12 = (x, x) YE V.
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.
Linear Algebra -- Inner Product Spaces Example 10. Use the inner product (u, v) = 2uivi + 3u2v2 for u = (1,3), and v = (2,4) to find the angle between u and v.
Let v and w be vectors in an inner product space V. Show that v is orthogonal to w if and only if ||v + w|| = ||v – w||.
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)...
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.
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) =...
1.(16) Let P be an inner product space with an inner product defined as <.g > Ox)g(x)dx a) Let / =1+x.8=-2+x-x. Compute: <.8 >. The angle between / and g, and proj, b) Let h=1+ mx' in P Find m such that and h are orthogonal c) Let B = (1+x.I-XX+X' is a basis for P. Use the Gram-Schmidt process to covert B to an orthogonal basis for P. 2. Suppose and ware vectors in an inner product space V...