(1 point Consider thetnction T.:K?K3 defned by T\"(z.xz)-(0,y,0). This kird offn

Question

(1 point Consider thetnction T.:K?K3 defned by T"(z.xz)-(0,y,0). This kird offndon is aled aprlection, since weare prleding'the vector (z, y,z)orto the aus. In this problem, you wil prove that the function T is linear. In the first part, you wil prove that T preserves addition. In the second part, you wil prove that T preserves scalar multiplication. There is only one comect answer for each part, so be sure not to skip any steps. 1. Use 7 of the sentence tragments to prove that T preserves aodition Choose from these Proof Then we have T((z 1, ,211+ (z2+Hz, 22)) = (0,n,0) + (0,.0) Let ,2 ,1h , gy, , be art trary elements of Then we have 2. Use 7 of the sentence tragments to prove that T preserves scalar muitiplication Choose from these Proof Then we have T(c(0,y,0)) Let be an arbitrary elment of K. and set z=: = 0. Let c be a scalar in K = Trez,cy,cz) Then we have T(c(r, y,:)) Let z, y,z and c be arbitrary elements of K e(0,y,0)

Explanation / Answer

1. i. Let x1,x2,y1,y2,z1,z2 be arbitrary elements of K.

ii. Then we have T((x1,y1,z1) +(x2,y2,z2))

iii. = T(x1+x2,y1+y2,z1+z2)

iv. = (0,y1+y2,0)

v. = (0,y1,0)+(0,y2,0)

vi. = T(x1,y1,z1) +T(x2,y2,z2)

vii. Therefore, T preserves addition.

2. i. Let x,y,z and c be arbitrary elements of K.

ii. Then we have T(c(x,y,z))

iii. = T(cx,cy,cz)

iv. = (0,cy,0)

v. = c(0,y,0)

vi. = cT(x,y,z)

vii. Therefore, T preserves scalar multiplication.

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