Show that the following functions define an inner product on (R)2 where U= (x,y) and V= (x2,y2) a) = 3x1x2 + y1y2 b) = (x1x2)/2 + (y1y2)/4

Q: Determine if the function defines an inner product on R2, where u = (U₁, U₂) and v= (v₁, v₂).…

A: Define a a function on R² by  ⟨ u , v ⟩ = 8u1v1+u2v1+u1v2+8u2v2 .   Now  Symmetric. ⟨ u , v ⟩ =…

Q: Determine if the function defines an inner product on R², where u = (u, v) = ₁V₁ satisfies (u, v) =…

A:

Q: Show that the function a Y1 (x, y) = [21 2] в с Y2 is an inner product on R? if and only if b2 – ac…

A:

Q: Describe how to find the Jacobian of x and y with respect to u and v for x = g(u, v) and y = h(u,…

A:

Q: 2) Use Green's Theorem to compute f F df where F= (3x +4y)i + 2xyj and C is the triangular path from…

A:

Q: Show that V spans P² where V = {1 + 2t + 2t², t + t², 1 − t²}

A:

Q: Find (u, v), ||u||, ||v||, and d(u, v) for the given inner product defined on Rn. U= (0, -3), v =…

A:

Q: Show that D^2 = {(x, y) ∈ E^2: x^2+y^2 ≤ 1} ⊂ E^2 and the space containing a single point are…

A: In algebraic topology, two topological spaces are said to be homotopy equivalent if there exists a…

Q: 9. Show that the function (-, -) : IR² × R² → IR defined by setting = #1#2 + 2x12 + 2y1*2 + 8y1Y2 /2…

A:

Q: Find (u, v), ||u||, ||v||, and d(u, v) for the given inner product defined on R". u = (5, 0), v =…

A:

Q: -2y 2x (x²+y²) ²³ (x²+y²)² Compute fF.dr, where F(x, y) = oriented counterclockwise. (Use symbolic…

A:

Q: \(d\) is a metric on \(X\). Show that \(\rho : X^2 \to [0,\infty)\) defined by \[ \rho(x,y) =…

A:

Q: Find (u, v), ||u|l, ||v||, and d(u, v) for the given inner product defined on R". u = (3, 0, 1), v =…

A:

Q: Use the functions f and g in C[-1, 1] to find (f, g), ||F||, ||9||, and d(f, g) for the inner…

A:

Q: Use the functions f and g in C[-1, 1] to find (f, g), ||f||, |lg||, and d(f, g) for the inner…

A:

Q: Let S = Þ (D), where D = {(u, v) : u² + v² ≤ 1, u ≥ 0, v ≥ 0} and Þ (u, v) = (2u+1, uv, 3u + v). (a)…

A:

Q: Let T: R3 -> R2 be defined by T(a, b, c) = (a + b, 2a —- c). DescribeT-1(1, 11).

A: T: R3 -> R2 Hence, T-1 : R2 ----> R3 So, necessarily we have to find a, b & c such that…

Q: True or False: (p, q) = p(1)q(1)+p(2)q(2) +p(3)q(3) defines an inner product on P2(R).

A:

Q: Justify that (•, ·) defines an inner product on R?. the function (', ·) defined on R? as follows: U1…

A:

Q: Use the functions f and g in C[-1, 1] to find (f, g), f, g, and d(f, g) for the inner product (f, g)…

A:

Q: Let ((uz,u2,U3),(V1,V2,V))= 2U1V1 + 3u2V2+ U3V3 be an inner product on R, Determine d(1, – 1, –…

A:

Q: Let V(F) be an inner product space and a, Be V. Show that a= B iff ( α, γ)- (β,)V γε V.

A:

Q: 6. Show that each of the following are not inner products on R³, (a) (u, v) = U₁V₁ + U2V2; (b) (u,…

A:

Q: Use the functions f and g in C[-1, 1] to find (f, g), ||f||, |lg||, and d(f, g) for the inner…

A:

Q: Q4) Consider the space (X.Tu) and let A = [0,2) U (3) U (4,7). Find d(A), A, Aº, ext(A) and b(A).

A:

Q: Explain why is not an inner product for u =(u1, u2) and v =(v1, v2) in R2 when = u1u2 + v1v2 using…

A:

Q: Given that x, y E R" and u · u is the ordinary dot product, justify that the function defined by (x,…

A:

Q: 1 (d) Last week, Amy walked a median of 7.5 kilometres each day. 1 kilometre = 0.6 miles The table…

A:

Q: Let θ : R^4 → R defined by θ(u) = sup{u1|, |u4|}. Determine if θ is a norm, a seminorm or none of…

A: The given function .Objective: To determine if  is a norm, a semi-norm or none of the two for

Q: 2. A robot performs the following operations on a set of points (x,y) in the 2D plane to check the…

A:

Q: Let f be a function that takes elements of R?rR? into R' as follows: f(u, v) = |u1v1|+ |u2v2|, where…

A:

Q: Show that H = y:x+2y-z + 1 = 0 is not a sub-space of R³. -1=0} is

A:

Q: f: X → Y and g: Y→ V be bijective functions. Show the following: ) gof: X → V is injective. gof: X→V…

A: It is given that f:X→Y and g:Y→V are bijective functions.