Homework SourseDetermine which of the following maps are linear If the map
Determine which of the following maps are linear. If the map is linear, then write down the matrix representation and determine if the map is bijective.
i) R : R3 -> R3: (x,y,z) -> (3x + 2y, -x + z, z2)
ii) S : R3 -> R3 : (x,y) -> (x - y, x + y + 1, x)
3? 1-2Solution
Let u = (x1, y1, z1) , v = (x2, y2, z2) be 2 arbitrary vectors in R3 and let c be an arbitrary scalar. Then:
(i) R(u+v) = f(x1 +x2 , y1 +y2 , z1 +z2) =(3x1+3x2 +2y1 +2y2, -x1-x2+z1+z2 , z1+z2 ) (3x1 + 2y1, -x1 + z1, z12) + (3x2 + 2y2, -x2 + z2, z22) as (z1+z2)2 z12 + z22. Hence R(u+v) R(u)+R(v). Thus, R does not preserve vector addition and therefore R is not a linear transformation.
(ii) S (u+v) = (x1+x2-y1-y2, x1+x2+y1+y2+1, x1+x2) (x1 – y1, x1 + y1 + 1, x1) +(x2 – y2, x2 + y2 + 1, x2) as x1+x2+y1+y2+1 ( x1 + y1 + 1)+( x2 + y2 + 1). Hence S(u+v) S(u)+S(v). Thus, S does not preserve vector addition and therefore S is not a linear transformation.
(iii) T(u+v) = (3x1+3x2+2y1+2y2-4z1-4z2, -x1/2 –x2/2+4y1 +4y2-5z1-5z2, -x1-x2+3y1/2+3y2/2+z1+z2) = (3x1 + 2y1 - 4z1, -x1/2+4y1 -5z1 , -x1+3y1/2+z1) +(3x2 + 2y2 - 4z2, -x2/2+4y2 -5z2 , -x2+3y2/2+z2)= T(u)+T(V). Thus T preserves vector addition. Also, T(cu)= T(c(x,y,z)) = T(cx+cy+cz) = (3cx+2cy–4cz, -cx/2+4cy -5cz, -cx+3cy/2+cz) = c(3x + 2y - 4z, -x/2+4y -5z , -x+3y/2+z)= cT(u). Thus T preserves scalar multiplication. Hence T is a linear transformation. The matrix representation of T is A =
3
2
-4
-1/2
4
-5
-1
3/2
1
The RREF of A is I3. Thus the columns of A are linearly independent. Hence T is injective. Further, sine the columns of A span R3, hence T is also surjective. Thus, T is bijective.
| 3 | 2 | -4 |
| -1/2 | 4 | -5 |
| -1 | 3/2 | 1 |
