State whether the following statements are true or false The

State whether the following statements are true or false: The function T : R^2 rightarrow R^3 given by T(x_1,x_2) = (2x_1 + 3x_2,4x_2 - 1 - x_1, x_1) is a linear transformation. If T : V rightarrow W be an isomorphism, then ker(T) = {0}. Every square matrix has a LU-decomposition. If.4 is an m Times n matrix, then A^TA is an m Times m matrix. In linear programming problems, all variables are restricted to positive values only. One of the quickest ways to plot a constraint is to find the two points where the constraint crosses the axes, and draw a straight line between these points.

Solution

a ) True

b ) True

d ) True

For example,

A = 1 2 of order 1 x 2

A^T = 1 ( order 2 x 1)

2

A^TA = 1 2

2 4

Hence ,

A^TA has of order 2x2

e ) True