Let V be a finite-dimensional inner product space over C and let u, w€ V be fixed non-zero vectors such that (u, w) 0. Let T:V→ V be the linear operator defined by T(v) = (v, u) w for all v € V. Find a formula for T similar to the one given for T. Show that T is self-adjoint if w = au where a € R.

Q: If u is a unit vector and a is a positive scalar satisfying au + = 0, find a and u.

A: Given: u is a unit vector and the scalar a is positive satisfying the equation au+1,-2,5=0 To find…

Q: A vector y in Rn can also be viewed as an n × 1matrix Y = (y). Show that ||Y||2 = ||y||2

A:

Q: Let V = R². For (u₁, U2), (v₁, v₂) € V and a ER define vector addition by (u₁, U₂) (V1₁, V₂) = = (U₁…

A:

Q: Let W1 and W2 be skew-symmetric tensors with axial vectors w1 and w2 respectively. Show that…

A: According to the given information,

Q: . Consider V1 1 and v3 = V2 = .0. 0 ) as vectors in R3. Let T: R3→R3 be a linear operator such that…

A: The solution is given as

Q: 2. Let R3 be a real vector space, and u = (u1, u2, u3), v = (V1, v2, v3) E R", then show that =…

A:

Q: Let V be a vector space of polynomials of degree 3. Let u = 4t3 + 8t² – 8t + 12, v = t³ + 3t² –t+4…

A: The vectors are linear independent if they cannot be written in form of a linear equation. The…

Q: Prove that Ov = 0 for any vector ve V. Make sure to justify all your steps!

A:

Q: Consider set V = { (x, y ), x,yER} with the following operations: addition (x1, y1) + (x2 , Y2) = (2…

A:

Q: If U and V are vector spaces, define the Cartesian product of U and V to be   U X V = {(u, v) : u is…

A: Given : U×V=u, v: u∈U, v∈V To Prove: U×V is a vector space.

Q: Which of the follwing vectors span R2 Let V be the Vector space R? 1.1], [-1,1]

A: Remark: A set of vectors S={s1, s2, s3,..., sn} is linearly independent if and only the equation…

Q: Prove that u is orthogonal to v - proju(v) for all vectors u and v in Rn, where u ≠ 0.

A:

Q: Let V be the space on R² where addition of the vectors u and v are defined as u+v= (2u1 +3v2, 3u2 —…

A: Here we have to find zero vector and inverse vector for given vector operation.

Q: H.W: Let V = R?, defined the addition by: (u1 +U2) + (V1 + v2)= ( U1 +V1 , U2 +V2 + 1) With standard…

A: step:-1 Vector space:- A vector space is a set V over a field F with two binary operation + and .…

Q: Let v=(1', 1, 3.), u=(0, -4', 1) are nonzero vectors in R³. Subtask (a). Whether a linear functional…

A: Functional Analysis:

Q: Let W1 and W2 be skew-symmetric tensors with axial vectors w1 and w2 respectively. Show that…

A: To prove this identity we will use an identity. Proof of that identity is given below:…

Q: In the space C[0, π] let f, g, h, and j be the vectors defined by f(x) = 1 g(x) = x h(x) = COS X…

A: In the space C[0, ],

Q: Let u, v, and w be vectors in an inner product space V and let c be a scalar   <u, cv>=…

A: Let u, v and w be vectors in an inner product space. Let c be a scalar. To show, <u, cv>=…

Q: Let vv be the set of all vectors of the form -a b vectors u and in R³ such that W = span{u, v}. ū =…

A: Given space W =

Q: Please solve the question with a good explanation. It is Linear Algebra

A: Given:

Q: Show that {t, e'}, which is a subset of F, the vector space containing all real-valued functions, is…

A:

Q: Let nonzero vectors ū,v,weR be linearly independent. Show that there does not exist a nonzero vector…

A:

Q: Let u and v be nonzero vectors in an inner product space V. Prove that u − projvu is orthogonal to…

A: As per the question, we aim to show that in an inner product space, the vector u - projv(u) is…

Q: Let u, v, and w be vectors in an inner product space V and let c be a scalar. Prove (u, 0) = (0, v)…

A:

Q: Show that V={(x,2020x): x in R} is a vector space

A: According to the given information, it is required to show that the given set is a vector space.

Q: Show that V = R2 is not a vector space over R with respect to the following operations: (a, b) + (c,…

A:

Q: Let V be the vector space of functions spanned by B = {b₁ = 6 sin x + 7 cos x, b₂ = 5 sin x + 3 cos…

A:

Q: : Let V be the set of all ordered pairs of real numbers (u1, u). Consider the following addition and…

A:

Q: Consider the operator S on the vector space R1[x] given by: S(a + bx) = -a + b + (a + 2b)x A) Find…

A: In this question, the basis of the operator S on the vector space R1x is to be determined.

Q: Let U be a vector space and u, v, w be three lincarly independent vectors in U. Is there a linear…

A:

Q: Let P, is the vector space of polynomials of the degrees2.Use coordinate vectors to that the…

A: The main objective is to check {p1,p2,p3} linear independent and basis of P2.

Q: Determine whether the following vectors in R' are linearly independent lineariy dependent: (i)…

A: As per our guidelines I am supposed to solve only one question. So kindly upload the other questions…

Q: Find the norm of x and the unit vector ü in the direction of x if ||| = u =

A:

Q: 3c

A: Step 1:a) To evaluate the adjoint (or conjugate transpose) of the linear operator T on the inner…

Q: im(T1 + 14T) im(14T† + T2).

A:

Q: Let R" be a real vector space, and u = (u1, u2, Un), v = (v1, V2, ..., Vn) E .... R", then show that…

A: For a real vector space, the inner product . , . satisfies the following properties a)  u+w , v =…

Q: Let V be the space on R² where addition of the vectors u and v are defined as u + v = (U₁ + V1,…

A: