Prove the following T is linear in the following definitions (a) T : R3 →R2 is...

Prove the following T is linear in the following definitions
(a) T : R3 →R2 is defined by T(x,y,z) = (x−y,2z)
(b) T : R2 →R3 is defined by T(x,y) = (x−y,0,2x+y)
(c) T : P2(R) → P3(R) is defined by T(f(x)) = xf(x)+f(x)

Expert Solution

Colby Messinger answered 7 months ago

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.

Let L be the set of all linear transforms from R3 to R2 (a) Verify that...

Let L be the set of all linear transforms from R3 to R2 (a) Verify that L is a vector space. (b) Determine the dimension of L and give a basis for L.

Let T: R2 -> R2 be a linear transformation defined by T(x1 , x2) = (x1...

Let T: R2 -> R2 be a linear transformation defined by T(x1 , x2) = (x1 + 2x2 , 2x1 + 4x2) a. Find the standard matrix of T. b. Find the ker(T) and nullity (T). c. Is T one-to-one? Explain.

(a) Find a linear transformation T : R2→R2 that (i) maps the x1-axis to itself, (ii)...

(a) Find a linear transformation T : R2→R2 that (i) maps the x1-axis to itself, (ii) maps the x2-axis to itself, and (iii) maps no other line through the origin to itself. For example, the negating function (n: R2→R2 defined by n(x) =−x) satisfies (i) and (ii), but not (iii). (b) The function that maps (x1, x2) to the perimeter of a rectangle with side lengths x1 and x2 is not a linear function. Why? For part (b) I can't...

The linear transformation is such that for any v in R2, T(v) = Av. a) Use...

The linear transformation is such that for any v in R2, T(v) = Av. a) Use this relation to find the image of the vectors v1 = [-3,2]T and v2 = [2,3]T. For the following transformations take k = 0.5 first then k = 3, T1(x,y) = (kx,y) T2(x,y) = (x,ky) T3(x,y) = (x+ky,y) T4(x,y) = (x,kx+y) For T5 take theta = (pi/4) and then theta = (pi/2) T5(x,y) = (cos(theta)x - sin(theta)y, sin(theta)x + cos(theta)y) b) Plot v1 and...

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)}

T::R2->R2, T(x1,x2) =(x-2y,2y-x). a) verify that this function is linear transformation. b)find the standard matrix for...

T::R2->R2, T(x1,x2) =(x-2y,2y-x). a) verify that this function is linear transformation. b)find the standard matrix for this linear transformation. Determine the ker(T) and the range(T). D) is this linear combo one to one? how about onto? what else could we possibly call it?

What is the arithmetic mean for the following stock returns: R1=6.3%, R2=4%, R3=7.2%?

You bought a bond for $950 1 year ago. You have received a coupon of $60. You can sell the bond for $977 today. What is your total dollar return?What is the arithmetic mean for the following stock returns: R1=6.3%, R2=4%, R3=7.2%?

Consider a transformation T : R2×2 → R2×2 such that T(M) = MT . This is...

Consider a transformation T : R2×2 → R2×2 such that T(M) = MT . This is infact a linear transformation. Based on this, justify if the following statements are true or not. (2) a) T ◦ T is the identity transformation. b) The kernel of T is the zero matrix. c) Range T = R2×2 d) T(M) =-M is impossible.

Assume that R1 = 44 Ω , R2 = 75 Ω , R3 = 19 Ω...

Assume that R1 = 44 Ω , R2 = 75 Ω , R3 = 19 Ω , R4 = 79 Ω , R5 = 20 Ω , and R6 = 23 Ω . fig 1 fig 2 fig 3 fig 4 Part A Find the equivalent resistance of the combination shown in the figure (Figure 1) . Express your answer using two significant figures. Req =______ Ω Part B Find the equivalent resistance of the combination shown in the figure...

Question