Determine if Linear [[x],[y]]=[[y],[x]]

Problem

T*[[x],[y]]=[[y],[x]]

Solution

1. Define the transformation as a function T mapping ℝ2 to ℝ2 where T(u)=[[(u_2)],[(u_1)]] for any vector u=[[(u_1)],[(u_2)]]

2. Check additivity by evaluating T*(u+v) for two vectors u=[[(u_1)],[(u_2)]] and v=[[(v_1)],[(v_2)]]

T*(u+v)=T*[[(u_1)+(v_1)],[(u_2)+(v_2)]]=[[(u_2)+(v_2)],[(u_1)+(v_1)]]

T(u)+T(v)=[[(u_2)],[(u_1)]]+[[(v_2)],[(v_1)]]=[[(u_2)+(v_2)],[(u_1)+(v_1)]]

T*(u+v)=T(u)+T(v)

3. Check homogeneity by evaluating T*(c*u) for a scalar c

T*(c*u)=T*[[c*(u_1)],[c*(u_2)]]=[[c*(u_2)],[c*(u_1)]]

c*T(u)=c*[[(u_2)],[(u_1)]]=[[c*(u_2)],[c*(u_1)]]

T*(c*u)=c*T(u)

4. Conclude that since both properties of linearity are satisfied, the transformation is linear.

Final Answer

T*[[x],[y]]=[[y],[x]]* is linear

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