7. If U¡¡ ‡ 0 are components of a tensor of the type (0, 2) and if the equation ƒU¡¡ + gU ji ³ = 0 holds ij w.r.t to a basis then prove that eitherf=g and Uij is skew-symmetric orf=-g and Uij is symmetric.

Q: (3) Suppose a vector x is orthogonal to vectors u and v. Show that x is orthogonal to u+v.

A: We will use the fact that x is orthogonal to u and v to prove that x is orthogonal to u+v also

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: Let V be a vector space, let c be a scalar and let u, v, w be vectors in V and ca scalar. Which of…

A:

Q: Let B=[u, be a base for V=R², where = (1,2) and =(-1,1). We know that every vector in V is a linear…

A:

Q: 4. If ij,k are mutually perpendicular unit vectors and u is an arbitrary vector, prove that (i-u)(i…

A:

Q: Prove that llu - vll >=llull -llvll for all vectors u and v in Rn.

A: According to the given information, it is required to prove the inequality.

Q: 2 Let L be the line in R³ that consists of all scalar multiples of the vector . Find the orthogonal…

A:

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

A:

Q: (2) For every vector u and vector v in se A) True B False

A: Introduction: Vector is a quantity that has both magnitude and direction. Some mathematical…

Q: If A and B are both second-order tensors and t, u, v, and w are all vectors, prove the following…

A: Given that if A and B are both second-order tensors and t, u, v, and w are all vectors.   (1). To…

Q: n Let ú and ú be two nonzero vectors in R" for which ú + 2v is orthogonal to ủ – 20. What is the…

A: We have to find the ratio

Q: (a) Suppose A=(S,+S) and B (S-S), Show that S = A + B A= A and B =-B (b) Suppose that A B, is a…

A:

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: 3- The scalar component of the vector F in the direction of K is (K.F А- .K, В- K C-

A:

Q: (1,1,..., 1)T be the column vector consisting of all 1s. Prove 0. 3. Let a = that La

A: Given: Let a=(1,1,......1)T  Then objective is to prove  La=0

Q: 7. Let S = Show that span S = R' and find a basis for R' consisting of vectors from S.

A: Span of vector : Let us consider a vector space V then the set S is aid to be the span of V if the…

Q: 2. Use abbreviated index notation to prove or disprove the following statements involving arbitrary…

A: Since there are multipart, so here find the solutions of first and second part.

Q: Let z, w E C. Clearly prove that z + w = z+ū. Let u, v and w be vectors in C". Clearly prove that (u…

A: a) Let z, w ∈ℂ. Prove that z+w=z+w. Let z=z1+iz2 and w=w1+iw2. Add the two complex numbers:…

Q: Does A(u x v) = Auxv +ux Av?

A:

Q: Q1. If O (0,0,0) in (x,y,z) 3-D space and r(2,5,1) draw the vector in 3D and find its direction.

A: Plot point in 3D

Q: Let the following vectors be in R³ a = (1,3,-1) b= (-2,1,3) c = (3,2,1) d = (1,-1,2) 16.- Obtain a…

A: Given vectors are a=(1,3,-1), b=(-2,1,3), c=(3,2,1), and d=(1,-1,2).

Q: Given K= (-2,-1), L=(2,4), M= (-3,-1) and N= (1,4), Prove that KĽ and MN are equivalent by Showing…

A: Two vectors are Equivalent if they have same magnitude and direction.

Q: Given ri = 3i - ej +In2 k, r2 = 5i + 7 e'j - 4 k and r3= 7i-7j+ k. Find the unit vector of (a)…

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Q: Find the new coordinate vector for the vector x after performing the specified change of basis. 10)…

A:

Q: 1. Consider the Cartesian tensors i) ii) Or Or.) (a) Identify their orders. (b) Rewrite them in…

A: Given the system

Q: Suppose M is the set of all real column vectors in R' whose components are equal, then the basis for…

A: See the detailed solution below.

Q: 18 f1. are the corresponding Christoffel symbols and are components of two symmetric covariant…

A:

Q: Use Exercise 47 to show that if cos α, cos β, and cos γ are the direction cosines of a vector v,…

A: cos2α,cos2β and cos2γ are the direction cosines of the vector v

Q: Let x and y be two orthogonal vectors. Then prove that +

A:

Q: Find a unit vector parallel to the vector a´. a'= (-2,3,–1)

A:

Q: Find the vector x determined by the given coordinate vector [x] B and the given basis B. 8) B =…

A:

Q: If T is a mixed tensor, then T} + T{ is generally not a tensor a tensor May or may not

A: Answer : may or may not  Hence, Option (iii) is true.