Find a matrix representation of transformation T(x)= 2x1w1+x2w2-3x3w3 from R3 to a vector space W, where...

Find a matrix representation of transformation T(x)= 2x1w1+x2w2-3x3w3 from R3 to a vector space W, where w1,w2, and w3 ∈ W. Clearly state how this matrix is representing the transformation.

Expert Solution

T:R³ → W is defined by T(X) = T(x₁,x₂,x₃)^T = 2x₁ w₁+x₂ w₂-3x₃ w₃, where w₁,w₂, and w₃ ∈ W.

Then T(e₁) = T(1,0,0)^T = 2w₁+0w₂+0w₃, T(e₂) = T(0,1,0)^T = 0w₁+1w₂+0w₃ and T(e₃) = T(0,0,1)^T = 0w₁+0w₂-3w₃. Hence, the standard matrix of T is A =

2w₁	0	0
0	w₂	0
0	0	-3w₃

Then T(X) = AX.

Let X =(x₁,x₂,x₃)^T and Y=(y₁,y₂,y₃)^T be 2 arbitrary vectors in R³ and let k be an arbitrary scalar. Then T(X+Y) = A(X+Y) = AX +AY = T(X) +T(Y). This implies that T preserves vector addition.

Also, T(kX)= A(kX) = kAX = kT(X). This implies that T preserves scalar multiplication also.

Thus, Tis a linear transformation.

If we need actual multiplication by A, then T(X+Y) = A(X+Y) = A(x₁+y₁,x₂+y₂,x₃+y₃)^T = 2(x₁+y₁)w₁+(x₂+y₂)w₂-3 (x₃+y₃)w₃ = 2x₁w₁+x₂w₂-3x₃₃+2y₁w₁+y₂w₂-3y₃w₃ = A(x₁,x₂,x₃)^T+A(y₁,y₂,y₃)^T = AX+AY = T(X)+T(Y). This implies that T preserves vector addition.

Also, T(kX) = A(kX) = A(kx_1k,x₂,kx₃)^T = 2kx₁w₁+kx₂w₂-3kx w₃ = k(2x₁ w₁+x₂ w₂-3x₃ w₃,) = kAX = kT(X). This implies that T preserves scalar multiplication also.

Hence , Tis a linear transformation.

milcah answered 3 years ago

Find the matrix A' for T relative to the basis B'. T: R3 → R3, T(x,...

Find the matrix A' for T relative to the basis B'. T: R3 → R3, T(x, y, z) = (y − z, x − z, x − y), B' = {(5, 0, −1), (−3, 2, −1), (4, −6, 5)}

Vector v=(9,0,2) is vector from R3 space. Consider standard inner product in R3. Let W be...

Vector v=(9,0,2) is vector from R3 space. Consider standard inner product in R3. Let W be a subspace in R3 span by u = (9,2,0) and w=(9/2,0,2). a) Does V belong to W? show explanation b) find orthonormal basis in W. Show work c) find projection of v onto W( he best approximation of v with elements of w) d) find the distance between projection and vector v

A linear transformation from R3-R4 with the V set of vectors x, where T(x)=0, is V...

A linear transformation from R3-R4 with the V set of vectors x, where T(x)=0, is V a subspace of R3?

Consider a transformation ?:ℝ3→ℝ3T:R3→R3 defined by ?(?,?,?)=(?+?+?, 2?+?, ?−2?+?).T(x,y,z)=(x+y+z, 2x+y, x−2y+z). (a) Find the standard matrix...

Consider a transformation ?:ℝ3→ℝ3T:R3→R3 defined by ?(?,?,?)=(?+?+?, 2?+?, ?−2?+?).T(x,y,z)=(x+y+z, 2x+y, x−2y+z). (a) Find the standard matrix of ?T. (b) Is ?T a linear transformation? Explain. (c) Is ?T invertible? (d) Find the image of (2,−1,1)(2,−1,1) under ?T.

In parts a and b, A is a matrix representation of a linear transformation in the...

In parts a and b, A is a matrix representation of a linear transformation in the standard basis. Find a matrix representation of the linear transformation in the new basis. show all steps. a. A = 2x2 matrix with the first row being 2 and -1, and the second row being 1 and 3; new basis = {<1, 2> , < 1, 1> } b. A = 3x3 matrix with the first row being 2, 1, -1, the second row...

Assume that a 3D polygon is tranformed by a transformation matrix T. Assume the normal vector...

Assume that a 3D polygon is tranformed by a transformation matrix T. Assume the normal vector of the polygon is n. Can we use the formula Tn to compute the normal vector of the transformed polygon? If not, drive the correct formula.

Consider the matrix transformation T in R2that does the following actions. It rotates a vector by...

Consider the matrix transformation T in R2that does the following actions. It rotates a vector by radians counterclockwise and then reflects the vector across the x axis. Apply T to i. Apply T to j. Apply T to another vector of your choice, but one that has magnitude 1 so it sits on the unit circle. What is the generalized form of T?

Let T : R2 → R3 be a linear transformation such that T( e⃗1 ) =...

Let T : R2 → R3 be a linear transformation such that T( e⃗1 ) = (2,3,-5) and T( e⃗2 ) = (-1,0,1). Determine the standard matrix of T. Calculate T( ⃗u ), the image of ⃗u=(4,2) under T. Suppose T(v⃗)=(3,2,2) for a certain v⃗ in R2 .Calculate the image of ⃗w=2⃗u−v⃗ . 4. Find a vector v⃗ inR2 that is mapped to ⃗0 in R3.

V is a subspace of inner-product space R3, generated by vector u =[1 1 2]T and...

V is a subspace of inner-product space R3, generated by vector u =[1 1 2]T and v =[ 2 2 3]T. T is transpose (1) Find its orthogonal complement space V┴ ; (2) Find the dimension of space W = V+ V┴; (3) Find the angle q between u and v; also the angle b between u and normalized x with respect to its 2-norm. (4) Considering v’ = av, a is a scaler, show the angle q’ between u...

Let T: V →W be a linear transformation from V to W. a) show that if...

Let T: V →W be a linear transformation from V to W. a) show that if T is injective and S is a linearly independent set of vectors in V, then T（S） is linearly independent. b) Show that if T is surjective and S spans V,then T（S） spans W. Please do clear handwriting!

Question