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Question

6) Show that vectors u1, u2 are linear independent if and only if u1 and u1 + u2 are linear independent.

(4) Show that super-positioning principle does not work for non-linear equations. (For example, yi 1, y2 -1 and y (y 1) (y -1).) (5) Show that super-positioning principle does not work for non-homogeneous equations. (For example, yi e2ar/3 and y (6) Show that vectors u1, u2 are linear independent if and only if u1 and u1 w2 are linear independent. (7) Show that e and are are linear independent TT TI (8) Following (2), show that e and are era are linear independent (9) Show that if f (r) and f2(z) are linear independent, then K1 fi(z) and K2 f2(z) are linear independent, for any nonzero constants K1 and K2

Solution

x) Suppose xxxx xxx xxxxxxx u1 xxx u2 xxx xxxxxxxy xxxxxxxxxxx .

xxxx means xxxx xxy xx their xxxxxx combination = x xxxxx the xxxxxxxxxxxx are zxxx. xx., xxxxx exists xxxxxxx a, x xxxx xxxx au1 + bu2 = x xxxxxxx a = b = x. xxx, we xxxx to xxxxx xxxx xx and xx + xx xxx xxxxxxxy independent.

xxxxxxx for xxy xxx xxxxxxx c, x we xxxx xxx + d(u1 + u2) = x .xx., (c+d)u1 + du2 = x. xxx since xy our xxxxxxxxxx xx, xx are xxxxxxxy independent, xx xxxx xxxx c+d =x and x = x

thus x = x xx xxxx

This xxxxx that xxx + x(xx+xx) = x implies x = x = x .

xxxx xx xxx u1+ xx are xxxxxxxy xxxxxxxxxxx xxxxxxxxxy assume xxxx u1and xx+xx xxx xxxxxxxy independent.

xxxx is xxxxx xxxxxx xxxxxxx m, x such xxxx xxx + n(u1 + u2) = x xxxxxxx m = n = x. xxx mu1 + n(u1+u2) = (x+x) xx + xxx

Since x = x, n = 0 , x + n xx also x. xxxx x+ n = 0 xxx x = 0, xxxxx are xxxxxxy xxx xxxxxxxxxxxx of xx and xx. xxxxx x, n , u1, xx xxx xxx arbitrary, xx follows xxxx xx xxx u2 xxx linearly xxxxxxxxxxx . xxxxx the xxxxx

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