Determine if Linear [[x],[y]]=[[0],[x]]

Problem

T*[[x],[y]]=[[0],[x]]

Solution

1. Define the transformation T:ℝ2→ℝ2 as T(u)=[[0],[(u_1)]] where u=[[(u_1)],[(u_2)]]

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

T*(u+v)=T*[[(u_1)+(v_1)],[(u_2)+(v_2)]]=[[0],[(u_1)+(v_1)]]

T(u)+T(v)=[[0],[(u_1)]]+[[0],[(v_1)]]=[[0],[(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)]]=[[0],[c*(u_1)]]

c*T(u)=c*[[0],[(u_1)]]=[[0],[c*(u_1)]]

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

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

Final Answer

T*[[x],[y]]=[[0],[x]]* is linear

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