3. A subspace U C Rº and x E R® are given by 3 1 -3 -1 -3 4 -3 2 U = 1 1 and x = 2 -1 2 2 1 7 2 [1/2 1 1/4 3/2] (a) Find the orthogonal projection of x onto U, TU (x). Please verify TU(x) lies in the same subspace as U. (b) What is your projection matrix?

Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider B = {1+x= 2x°, -1+a…

A: given B=1+x-2x2 , -1+x+x2 , 1-x+x2B'=1-3x+x2, 1-3x-2x2 , 1-2x+3x2 b) find the coordination matrices…

Q: Find a basis for R³ relative to which the matrix for T is diagonal, where T(x₁, x2, x3) = (4x₁ + x3,…

A:

Q: Find the projection matrix P for the projection onto C(A).

A: Given,               A=-21-1-11-2, and b→=3-60.

Q: Prove that the vectors (1,1,0), (3,1,3) and (5,3,3) are linearly dependent.

A:

Q: Let V₁, V2, and V3 be vector spaces and T₁ : V₁ → V₂ and T2 : V2 → V3 be linear transforma- tions.…

A:

Q: What is the matrix P = (P₁) for the projection of R³ onto the subspace V spanned by the vectors a₁ =…

A:

Q: Determine a vector that is orthogonal to both (1, 4, −1) and (5, 1, 0).

A:

Q: Find all vectors in R' that are perpendicular to both [1,0, 1, 0] and [1,2,0, 1] and then find a…

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: Let fi = 1+a + 5a², f2 = 1 + 2x + 8a². B = (f1, f2) is a basis for the subspace V = {ao +a;# + azw²…

A: s = - 10 (option a) The detailed solution is as follows below:

Q: Show that any isometry of R^n which fixes the origin is linear, and hence must be of the form f(x) =…

A: Introduction A square matrix is said to be an orthogonal matrix 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 vector equation for the line through the point (4, 5, -1) perpendicular to the vectors , 6,…

A: we have to find the line equation passing through point (4,5,-1) and perpendicular to the vectors…

Q: What is the length of EF? y 4 8 units 10- 7 units 4 2 units -10 -8 -6 4 10x -2- 6 units -4- -6 -8

A: solution of the given problem is below ...

Q: 6. Let V be the vector space of all diagonal 5 x 5 matrices, then dim(V)=--

A: Given that the vector space contains all diagonal 5×5 matrices and we need to find its dimension.

Q: (1 point) Express the vector = 8 as a sum of the basis vectors x = and y = %3D -13 xt j. = Netes V

A:

Q: V₁ = V1 V3 = V2= 4 [ 3 [-1] 2 3]

A: We use the properties of linear combination to

Q: a) Let t: R">R be the Composite 3. linear transformation which means the vectors in R? are first…

A: Answer:

Q: The matrix that projects onto the x-y plane is

A: We have to find the matrix that project onto  plane

Q: {} Find in R² whose coordinate vector relative to the basis B is [x]B Let B = x = 12 -5 -6 [:] 5

A: Introduction: The coordinate vector of an element of vector space related to basis is the vector…

Q: The equatiokn  Y = (x,y)T  is given as a vectorfunction with 2 components and A = table below 3 2 2…

A: Given is a matrix A and we have to diagonalize the matrix A. First we find eigenvalue of matrix A…

Q: Find the change-of-coordinate matrix P from B to C and P from C to B for the bases B = {1,t,t^2} and…

A: Let U={u1,u2,⋯,un} and V={v1,v2,⋯,vn} are two bases of a vector space W.The identity map I:W→W…

Q: The cross product of two vectors in R is defined by azb3 – azb2 [b1 x b2 a1 a2 azbı – ajb3 - a3 a,b2…

A:

Q: Determine which set of vectors is linearly dependent (a) (b) , (c) (d) (e) (f) O (a) O (b) O (c) O…

A: Recall:A set of vectors is said to be linearly dependent if the determinant is zero.If the…

Q: Find all vectors u such that ux(1,1,1)=(8,-5,-3).

A:

Q: Let B = (1, x, x², x³) be an ordered basis for P3. Find the coordinate vector 8x³ + 2x²7x - 9…

A:

Q: d. Find the transition matrix from C to B. e. Find the coordinates of u = (2, –2) in the ordered…

A: