In the vector space R the vectors a = (1,0,0,-1), 5= (1,0,1,2), y = (1,0, 8, 1) are given. (a) Is {a, 3,y} linearly independent? (b) Find an orthogonal basis for the subspace W = with respect to the standard inner product. (c) Find the orthogonal projection of the vector (1, 1, 1, 1) onto the sub- space W.

Q: Find the coordinates of the vector v = (7,4) with respect to the basis U₁ = (2, 1) and u₂ = (3,1).

A: The answer is given below:

Q: What is the scalar projection of vector a given by (a₁,a2,a3) in the direction of vector b given by…

A: We have to find the scalar projection of a in the direction of b.

Q: [2° [3] 2) w1 2, w2 = |2 ve w3 = |4 vektörlerinin birleşimiyle oluşturulan üçgenin alanını,…

A:

Q: (iii) {(x, y, z) = R³: (x, y, z). (1, 1, 1) = 0} = {(x, y, z) = R³ : x+y+z = 0} = {(1,-1,0), (1, 0,…

A: Given,   {(x,y,z) ∈ R3:(x,y,z)-(1,1,1)=0}={(x,y,z) ∈ R³:x+y+z=0}=L(1,-1,0),(1,0,-1)} and we need to…

Q: Find a basis for the subspace of R° that is spanned by the vectors V1 = (1, 0, 0), v2 = (1, 1, O),…

A:

Q: Feristic values and the correspondi @quation (A -XB) X=0 if & B = aty 1 V= 0 04 relations and 1 2 3…

A:

Q: What is the matrix representing a translation by vector [−1, 2, −1]? What is the matrix representing…

A: The given matrix is -12-1. The rotation angle at 90°. So if the rotation is θ around Y axis is,…

Q: Prove that the vectors (a1, a2), (B₁, B₂) = R× Rare linearly dependent iff α₁ß2 - α2ß₁ = 0.

A: The given vectors are  and We have to show that the given vector is linearly dependent if

Q: Show linear dependence in part (ii) to show that ({1,0,1}) is a basis.

A: From part ii)We have to show that the set  is a basis of .We know thatA basis B of a vector space V…

Q: Find a basis for the subspace of R° that is spanned by the vectors V1 = (1, 0, 0), v2 = (1, 1, O),…

A:

Q: The two vectors u=(-2,3,1,4) and v= (1,2,0, -1) are orthogonal in the vector space R4 True False

A:

Q: Find a basis for the subspace of R° that is spanned by the vectors V1 = (1, 0, 0), v2 = (1, 1, O),…

A: We have to find a basis for the subspace of R³ that is spanned by the given vectors.

Q: Find the terminal point Q of the vector v = 7i-j+3k given the initial point is P(-2, 3, 5) О (5, 2,…

A: In this question we have to find the terminal point of Q.

Q: Q-2: a) combination of the vectors X = (1,2,3,0) and Y = (2,1,6,0). b) a linearly independent set of…

A:

Q: 3. Let W C R be an inner product subspace with standard dot product. { 12 then If W I1- 2r2 + 4 = I3…

A:

Q: Apply Gram-Schmidt orthogonalization to the column vectors of A, a1 = (1, 1, 0)T ,a2 = (1, 0, 1)T ;…

A:

Q: c) Find dimension of subspace of all vectors in R° whose first and third entries are equal.

A: We have to find the dimensions

Q: Determine the vector projection of the vector (5,0, 7) on the vector (–1,3,4).

A:

Q: Use matrix multiplication to find the orthogonal projection of (-8, 5, 7) on the xy-plane. Let T…

A:

Q: Let B' = {u1,u2} and ß = {v1,v2} be ordered bases for a 2-dimensional vector space V. Given that [u1…

A: Given u1+u2β=5,4t, and the change of coordinate matrix Q from β' to β is Q=1c3d. Express v1 as a…

Q: 6. Project the vector b = (1, 4, 2) onto the plane S that contains the vectors a₁ = (1, 2, 1) and a2…

A:

Q: Construct an orthonormal set of vectors in ℝ3 from the set of linearly independent vectors ? =…

A:

Q: Let the following vectors be in R³ a = (1,3,-1) b = (-2,1,3) c = (3,2,1) d= (1,-1,2) 11.- Define the…

A: The given vectors .The aim is to find the norm of the vector .

Q: Find a basis for the subspace of R° that is spanned by the vectors V1 = (1, 0, 0), v2 = (1, 0, 1),…

A: Solution is given below