Let T : R3 → R3 be a linear transformation that reflects each vector x=(x1,x2,x3) through the plane x3=0 onto T(x)=(x1,x2,−x3). Show that T is a linear transformation.

Question

Nicole Conner · Accepted Answer

Step 1 A transformation T:Rn→Rn is said to be linear transformation if it satisfies the following conditions. T(u+v)=T(u)T(v) T(cu)=cT(u) where u and v are vectors in Rn and c is any scalar. Step 2 It is given that x=(x1,x2,x3) and T(x)=(x1,x2,−x3) Check whether T(x+y)=T(x)T(y) Let x=(x1,x2,x3) and y=(y1,y2,y3), such that x+y=(x1+y1,x2+y2,x3+y3). Consider the LHS, T(x+y)=(x1+y1,x2+y2,−(x3+y3)) =(x1+y1,x2+y2,−x3+y3) =(x1,x2)(y1,y2−y3) =T(x)T(x) =RHS Thus, it satisfies the first condition of linear transformation. Step 3 Similarly, check whether T(cx)=cT(x) Let c be any scalar, and x=(x1,x2,x3). Consider the LHS, T(cx)=T(cx1,cx2,cx3) =(cx1,cx2,−cx3) =cT(x) =RHS It satisfies the second condition of linear transformation. Step 4 Thus, T is a linear transformation.

Let T : R^3 \rightarrow R^3 be a linear transformation that reflects each

Answered question

Answer & Explanation

New Questions in Other