Is T: M22 --> R a Linear Transformation?

[FourierX]

Linear Transormation?

Homework Statement

Is function T a linear transformation for T: M₂₂ --> R by

T [a b]
[c d] ( i.e. T [a b ; c d] )= a + b + c + d

Homework Equations

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

T(ru) = rT(u)

The Attempt at a Solution

The matrix form is confusing me

 

[VeeEight]

Let U be the matrix you defined and let r be an arbitrary scalar. You know how to calculate rU and you have the formula for calculating T(U), so does T(rU) = rT(U)?

For the next part, define another matrix similarily as you did except with different letters. Call it V. Does T(U + V) = T(U) + T(V)?

 

[FourierX]

the first part T(rU) = rT(U) satisfies

 

[FourierX]

and i so does the second part! Am I right?

 

[VeeEight]

If you can show it then you are done.