Let x, y, and z be vectors in Cn and let a and B be complex scalars. Show that a ( z, x) +B ( z,y ) ( z, ax + By >

Q: Find the scalar product a•b when (i) a= x+2y -z and b=2x+3y +z (ii) a=x-3y +z and b=2x+y +z

A: We have to find the scalar product a·b when (i). a=x+2y-z and b=2x+3y+z(ii). a=x-3y+z and b=2x+y+z

Q: (a) Prove that if u is orthogonal to both v and w, then u is orthogonal to v + w. (b) Prove that if…

A: Given: (a) If u is orthogonal to both v and w. To prove:  u is orthogonal to v + w. (b) If u is…

Q: If x, y and z are vectors in an inner product space such that (x, y) = (x, z), then y = 2.

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: A, B, and C are 3 distinct points with non-zero position vectors a, b, and e respectively. a If a x…

A: Given three distinct points are A, B, C. with position vectors a, b, c respectively.…

Q: Calculate the scalar triple product u · (v × w), where u =〈1, 1, 0〉, v =〈3, −2, 2〉, and w =〈4, −1,…

A: The vectors are, u=1,1,0v=3,-2,2w=4,-1,2 We need to find the scalar triple product u·v×w

Q: Let x and y be vectors in the R³. Suppose that x is a scalar product of 2 and the vector y. Prove…

A: x and y are vector in R3, x is scalar or dot product of 2 and y vector,  it means x = 2.y,

Q: Show that (x, y) = 5x1y1 – 9x2y2 for vectors x = (x1, x2) and y = (y1, y2) in R2 defines an inner…

A: This question is about application of linear algebra

Q: Suppose u, v and w are nonzero vectors in 3-space and a is a scalar. Which of the following are…

A: We have to choose the correct answer from the given options

Q: If F(x,y,z)=1i+2j+3k, then O1. P=1 Q=2 and R=3 are vectors of F. O II, P=1 Q=2 and R=3 are gradient…

A:

Q: Find all the vectors of the family (2-jx, j4, 2-j5, jx) that are of hermitian length equal to 21.

A: Let Y=2-jx,j4,2-j5,jx Length =Y=4+x2+42+4+25+x2   Y=49+2x2 49+2x2=2149+2x2=2122x2=392x=±14

Q: Suppose u, v and w are nonzero vectors in 3-space and a is a scalar. Which of the following are…

A:

Q: 2- Determine a scalar x so that the given vectors are orthogonal. 1. A = 2i -xj +3k B = 3i +2j +4k…

A:

Q: The scalar product 0v is 0.

A:

Q: a. The zero vector 0 is in ker(A) b. If x and y are in ker(A), then so is z = x + y. c. If x is in…

A:

Q: Let V=R^2 and let H be the subset of V of all points on the line 2x+3y=6. does H contain the zero…

A: Given  V=R2 and let H be the subset of V of all points on the line 2x+3y=6.

Q: . Show that (x, y) = 4x1y1 – 7x2y2 for vectors x = (x1, x2) and y = (y1 y2) in R2 defines an inner…

A: Given two vectors x and y in R2. We need to show that the given function in x and y defines an inner…

Q: Let S be the space of all complex vectors (x, y, z) such that y = z². Determine whether or not S is…

A: In the given problem we verify the properties of vector addition

Q: Consider vectors a = (2, -8), b = (-1, 4), andc = 0. Determine the non-zero scalars a and B such…

A: Consider vectors a=(2,-8) , b=(-1,4) and c=0. We have to determine the non-zero scalars alpha and…

Q: Let v1 , v2 , . . . , vn be linearly independent vectors in a vector space V. Show that v2, . . . ,…

A: If the given vectors are linearly independent in the vector space V. Then it follows that

Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B' A) Given the…

A:

Q: Let u = 10i + 2j, v=i+j, and w = i-j. Find scalars a and b such that u = av + bw. a = b =

A:

Q: Determine a scalar r such that Ax = rx, where 1 A =| 1 1 and x = 4 -4 5

A: Ax=r x ⇒ x is an eigenvector of Afor the eigenvalue r. So to find r, we compute all the eigenvalues…

Q: Consider the set V of all real triples of the form (a, b, c) where c = a - 2b under the usual vector…

A:

Q: Find the dimension of the vector space.R6

A:

Q: let v =〈−2, 5〉and w =〈3, −2〉. Find a scalar α such that ||v + αw|| = 6.

A: Given, v=-2, 5w=3, -22

Q: 13. Recall that nonzero vectors u and are called collinear if there exists = a nonzero scalar a such…

A: Given data:- Nonzero vectors are u→ & v→ Nonzero scalar α v→=αu→

Q: Find the scalar and vector projections of b onto a. a = (4,7,-4) b =(4,-1, 1)

A:

Q: Determine a scalar r such that Ax = rx, where

A: Matrix: A=12-11014-45x=-12141AX=12-11014-45.-12141

Q: Let u = 7i- 3j, v= i-j, and w= i+j. Find scalars a and b such that u = av + bw. a = b=

A: We will find a and b

Q: 10. Show that V = R² with the standard scalar multiplication, but addition defined by (7, , v.) +…

A:

Q: Determine a scalar s such that A2 x = sx when Ax = r x.

A: Given that Ax=rx. Multiply both sides of the above equation by A and simplify as follows.

Q: Let u = 2i - j, v = 3i + j, and w = i + 4j.Find specified scalar (4u) . v.

A: The vector are given as, u=2i-j , v=3i+j and w=i+4j. To find: the value of 4u.v Solution: The value…

Q: Let u = 2i + j, v = i + j, and w = i - j. Find scalars a and b such that u = av +bw.

A: Given u = 2i + j, v = i + j, and w = i - j.Plug them in the equation u = av +bw.

Q: Determine a scalar r such that Ax = rx, where 1 A = and X=

A: Given, A=2112, x=11

Q: Let H be the set of all points in the second quadrant in the plane V=R^2. That is, H={(x,y) | x ≤ 0,…

A: Here in the question H be the set of all points in the second quadrant in the plane V=R2. That is,…

Q: Find the scalar and vector projections of b onto a. a = 3i – j + 4k, b = j+ ÷k

A:

Q: Determine a scalar x so that the given vectors are orthogonal. A = xi +-j +xk 2. B =-3i +4j + xk

A:

Q: If Aj is a covariant tensor of the second order and Bi, Ci are contravariant vectors; prove that…

A: Given: Aij is a covariant tensor of 2ndorder To find: AijBiCj is an invariant tensor

Q: Let 12, v, w be vectors In an inner product space such that //u|| =5, ||v||=5, and (u, v) =-2. Use…

A:

Q: Find the scalar AND vector projections of a onto b where a=(0,1,-2) and b=(-1,0,3)

A: Given , a = (0,1,-2) and b = (-1,0,3).

Q: Show that if k is a scalar and A is n × n, then det(kA) = kn det(A).

A: Show that if k is a scalar and A is n × n, then det(kA) = kn det(A). For any scalar k, we need to…

Q: Determine a scalar x so that the given vectors are orthogonal. A = 2i – xj +3k B = 3i +2j +4k

A:

Q: Suppose that u, u and w are vectors in a real inner product space such that (u, w) = 16, (v, w) = 5,…

A: Instead of u¯,v¯,w¯ , we are writing u,v,w. (for sake of convenience). Given: (i) <u,w>=16(ii)…

Q: Let x, y, and z be vectors in Cn and let α and β becomplex scalars. Show thatz, αx + βy} = α z, x} +…

A: Given: z,αx+βy=αz,x+βz,y

Q: Show that (x, y) = 4x1y1-7x2Y2 for vectors x = (x1, x2) and y = (V1 Y2) in R2 defines an inner…

A:

Q: Suppose that two vectors T and ỹ in R" are orthogonal, meaning their inner product satisfies ·g = 0.…

A: This is a problem of vector calculus.

Q: Determine a scalar r such that Ax = rx, where

A: