Hi guys, I need help in these two problems that I have no clue on how to begin. I would really appreciate it if you could help me out. Thanks.
1) Consider the homogeneous system AX = 0 having m equations and n variables.
a/ Prove that, if X1 and X2 are both solutions to this system, then X1 + X2 and any scalar multiple cX1 are also solutions.
b/ Consider a nonhomogeneous system AX = B having the same coefficient matrix as the homogeneous system AX = 0. Prove that, if X1 is a solution of AX = B and if X2 is a solution of AX = 0, then X1 + X2 is also a solution of AX = B.
(the "1" and "2" next to X are subscripts.)
2) Prove that the following homogeneous system has a nontrivial solution if and only if ad - bc = 0:
ax1 + bx2 = 0
cx1 + dx2 = 0
(the "1" and "2" next to x are subscripts.)
1) Consider the homogeneous system AX = 0 having m equations and n variables.
a/ Prove that, if X1 and X2 are both solutions to this system, then X1 + X2 and any scalar multiple cX1 are also solutions.
b/ Consider a nonhomogeneous system AX = B having the same coefficient matrix as the homogeneous system AX = 0. Prove that, if X1 is a solution of AX = B and if X2 is a solution of AX = 0, then X1 + X2 is also a solution of AX = B.
(the "1" and "2" next to X are subscripts.)
2) Prove that the following homogeneous system has a nontrivial solution if and only if ad - bc = 0:
ax1 + bx2 = 0
cx1 + dx2 = 0
(the "1" and "2" next to x are subscripts.)