Determine if the given function is a linear transformation T

Determine if the given function is a linear transformation. T(x_1, x_2) = (7x_1 + x_2, -5x_1 + 8x_2) The function is a linear transformation. The function is not a linear transformation. If so, identify the matrix A such that T(x) = Ax. (If the function is not a linear transformation, enter DNE into any cell.)

Solution

T(x) = Ax

x = [ x1, x2 ]

T(x) ={ 7x1 + x2 , -5x1+ 8x2 }

Let A = [ a b , c d]

Ax= T(x1, x2)

[ ax1 +bx2 , cx1 +dx2 ] = { 7x1 + x2 , -5x1+ 8x2 }

comparing with RHS

we get a = 7 ; b = 1 ; c = -5 ; d=8

A = [ 7 1 , -5 8 ]