To determine: To prove: The function T:V→W is linear transformation if and only if T(au + bv)=aT(u) + bT(v) for all vectors u and v and all scalars a and b.

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To determine:  To prove:  The function  T:V→W is linear transformation if and only if  T(au + bv)=aT(u) + bT(v) for all vectors u and v and all scalars a and b.

broliY · Accepted Answer

Approach:  The function T is said to be linear transformation if is satisfy the vector addition and scalar multiplication properties.  The linear transformation is given by,   T(au + bv)=aT(u) + bT(v) … (1)  The vector addition is given by,  T(u + v)=T(u) + T(v) … (2)  The scalar multiplication is given by,  T(cu)=cT(u) … (3)  Calculation:  Let a=1andb=1.  Substitute 1 for a and b in formula (1).  T(1u + 1v)=1T(u) + 1T(v)  =T(u) + T(v)  This shows that the function T is preserves T in vector addition.  Substitute au for cu in formula (3)  T(au)=bT(v)  Substitute au for cu in formula (3)  T(bv)=bT(v)  This shows that the function T is preserves in scalar multiplication.  From above calculation it is clear that the function T preserve the vector addition and scalar multiplication. So, the function T is a linear transformation.  Conclusion:  Thus, the function T:V→W is a linear transformation if and only if  T(au + bv)=aT(u) + bT(v)  for all vectors u and v and all scalars a and b.

To determine: To prove: The function T:V rightarrow W is linear transformation if and only if T(au + bv)=aT(u) + bT(v) for all vectors u and v and all scalars a and b.

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