Let T: IR2 > R2 reflection (or orthogonal symmetry) with respect to the line 2x - y = 0. We call back that the vector a-(-) is orthogonal to the line in question and the vector b = (;) is on this line, a) Give T (a) and T (b) (think about the geometry of reflection) b) Give the canonical matrix of T. Hint: use a) and an inverse matrix and a c) Express the vector e = (6) as a linear combination of a and b and using linearity of T, give T (e1). Do the same with e- () d) Using c), find the canonical matrix of T and give the vector obtained by the reflection of the vector (3) in relation to the line in question.

Q: Find a transformation matrix to rotate a vector in R2 through an angle of 30 degrees. Then use it to…

A: According to the given information, it is required to find a transformation to rotate a vector in R2…

Q: Determine whether the following matrix is orthogonal or not: cosa sina A = sinßsina cosß -sinßcosa…

A: To determine if a matrix is orthogonal or not, multiply the matrix by its transpose. If the…

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.- Consider the following two vectors, vi = and v2 = Do you consider vector vz a rotation or…

A:

Q: Determine whether the vectors (3,0, 3, 6), (6,0, 0, 3), (0,0,3,0),(6,3,9,12) form a basis of R*.

A:

Q: Write down the 4 x 4 rotation matrix that takes the orthonormal 3D vectors u (Xu, Yu, Zu), V = (xv,…

A:

Q: Let B = matrix from B to C 8 - { }], ] } and c-{[7], [43]} C = be two bases for R². Find the change…

A:

Q: "Write x as the sum of two vectors, one in Span (u,.u₂.3), and one in Span (U4). Assume that (u) is…

A: In this question , we will write the 'x' as linear combination of remaining vectors .

Q: Let B = {(1, 1, 1), (1, 5, -3), (2, 2, 1)}. Show that B is a basis for R'. For the given %3D vector…

A: To solve the following, we use the standered result of basis and spanning set Basis- Let V is any…

Q: Evaluate the Dot Product u * (v + w). If : u = (-5, 7) v = (1, 6) w = (-11, 2)

A:

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: Show that the vectors a, (1, 0, 0), a2 = (1, 1,0), a, = (1, 1, 1) form a basis of C'(C).

A:

Q: Let a= and b= Show that there are scalars s and t so that sa+tb=. s = ? t = ?

A: a=-5,0,-3 and b=4,0,2sa+tb=27,0,15

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: Find an orthogonal basis for A1, A2, A3 (i.e., find 3 mutually perpendicular vectors u1, u2, u3 with…

A: The solution are next step

Q: Select all the functions that are a solution of y" - y = 0

A:

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

A:

Q: With each m xn matrix Y we associate the column vector vec(Y) of length m x n defined by vec(Y) :=…

A:

Q: Let L be the line that passes through the origin in the direction of u = (1, 1, 1). Find the matrix…

A: Let L be the line that passes through the origin in the direction of u = (1, 1, 1) The matrix for…

Q: Using the transformation law for vectors, prove that for any two vectors Ā, B, the inner product of…

A: As per the question we have to prove that the inner product of two vectors is invariant of…

Q: Let T be the reflection about the line -2x + ly = 0 and T, be the reflection about the line 2x +…

A: Every matrix has 2 attributes, rows and columns, if a matrix A is represented as Amxn, then m…

Q: Consider linear mapping A E L(Vn) and two basis X, Y of Vn. Select all true statements. Let Y be a…

A: Given that, Vn→vector space X=x1, x2, … , xn; Y=y1, y2, … , yn bases of Vn. A∈LV Verify the given…

Q: Derive the composite matrix for rotation about arbitrary point (a,b) in clockwise direction  with…

A:

Q: Let T be the flection about the line -2x + ly = 0 and T, be the reflection about the line 2x + ly =…

A: The matrix of reflection about a line y=mx is. A=11+m21-m22m2mm2-1 Given that T1 is a reflection…

Q: Find the standard matrix of the linear operator M : R R' that first project every vector through…

A: We have to find Standard Matrix of Linear Operator  M : R3 -> R3

Q: r projection oE the following

A: we've to show whether the projection operators are positive operators or not. Now a projection…

Q: Based on Figure 2 below, explain the process if we want to transform the cow as in figure (with…

A: Assume that the cow is standing on a vector A→. We need to use suitable transformations, to make…

Q: Show that the vectors (4-3) and (5,2) form a basis CF TR

A:

Q: Compute the standard matrix of the linear operator M : R³ → R° that first reflects all vectors about…

A: