Given y € R", define Sy : R" → R by Sy(x) = x y for all x € R" (where Problem 2. x. y is the dot product of x and y). a. Show that Sy: R → R is a linear transformation for any y ER". b. Show that every linear transformation T: R" → R arises in this way; that is, T = Sy for some y € R".

Q: 1. If T(x,x,,x,) =(4x, – 2x, +5x,,x, - 7x,,6x, – 3x, – x,) is a linear transformation from ~3, find…

A:

Q: 11. A linear transformation T: R2 → R² first reflects points through the ₁-axis and then reflects…

A:

Q: 8. Let T: M₂2 → R² be defined as 7 ([a b]) = (a − b, cd). Determine whether or not 7 is a linear…

A:

Q: 4. Determine whether the mapping S: M2(R) → M2(R) defined by S(A) = A + AT is a linear…

A:

Q: 8. Let T: R³ → R² be a linear transformation and suppose T(0, 0, 1) = (2, 1), T(0, 1, 1) = (0,5),…

A: Let  be a linear transformation and suppose  and . The objective is to determine..

Q: Let A : R° → R³ and B : R → R´be two linear transformations. Then O A is onto and B is one-to-one. O…

A:

Q: x 5. T| y N x-y + z 3z] [2x + y

A: To prove that the given transformation is a linear transformation. Txyz=x-y+z2x+y-3z

Q: 34. Let T: R² → R² be the function that maps each point in R² to its reflection with respect to the…

A:

Q: Let T : R3 → R² be a linear transformation. Find, [8] T3 if T0 1 and T 1 =[] [3 %3D

A: See the detailed solution below.

Q: Problem 3. Let A be the linear transformation from R" to Rm. Define AC = {Ax|x = C'} and A-¹D = {x|…

A: Solution:  We  have  to  first  recall  the  definition  of  a  convex set.Suppose  A⊂ℝn. Then, A…

Q: 1. Let TR → Rm be a linear transformation, and let {V₁, V2, V3} be a linearly dependent set in Rn.…

A:

Q: 6.) Prove that the composition of two linear transformation is linear; that is, prove that if F:V→U…

A: Here, given that F:V→U, G:U→W both are linear transformations. To prove (F∘G)(u)=F(G(u)), where u∈U…

Q: 4. Determine a generating set for the range of the linear transformation T: R3 → R3 X1 - X2 + X3…

A: Given linear transformation:T:ℝ3→ℝ3 Tx1x2x3=x1-x2+x3-x1+2x2+x32x1-x2+3x3 We need to determine a…

Q: Let L : R → R³ be a linear transformation defined by L x2 x3 What is the range of L? t, s eR 0. teR…

A: Basis of the linear transformation

Q: 1 3 Let T: R* → R' be a linear transformation. Find T if -2 -3 2 1 T andT 1 3 -1 1 1

A:

Q: Let T: R2 → P, be a linear transformation for which TH= 1- 2x and = x + 2x?. 1 a and T 9. Find T

A: We use properties of linear transformation here.

Q: 3. Consider the linear transformations T: R2 R2 and S: R2 R2 defined by X2 ¹(E)-E) (C)-Ld 3x₁ Find…

A:

Q: Let T be a linear transformation from R" to R$ . Determine whether or not T is one-to-one in each of…

A:

Q: 1. Assume that T is a linear transformation. Find the standard matrix of T. (a) T:R? → R' with T(e1)…

A: