If T Rn rightarrow Rm is a linear transformation then T0 0

If T: R^n rightarrow R^m is a linear transformation then T(0) = 0 The transformation T: R^m rightarrow R^n given by T(x) = Ax is one to one if the matrix A has a pivot in every row. If a is an m times matrix whose columns do not span R^m, then the equation Ax = b is inconsistent for some b in R^m. If v_1, v_2, v_3, v_4 are 4 vectors in R^3 then {v_1, v_2, v_3, v_4} are linearly dependent.

Solution

All are true .

(a). If T(0) is not equal to 0 then it is not an L.T.

(b). T(x) = Ax is a equation of a line. So it always one to one.

( c). It\'s a theorem.

(d). Four vectors in 3D never be linearly independent.So they are always linearly dependent.