Which of the following sets are subspaces of R3 1 point Whi

Which of the following sets are subspaces of R3 ?

(1 point) Which of the following sets are subspaces of R3?

A. {(x,y,z) | x<y<z}
B. {(5x9y,6x7y,3x+7y) | x,y arbitrary numbers }
C. {(x,y,z) | x+y+z=0}
D. {(x,y,z) | 2x8y=0,4x+5z=0}
E. {(2,y,z) | y,z arbitrary numbers }
F. {(x,y,z) | x+y+z=5}

Solution

C. This is a subspace. As we can see here that it contains the zero vector.
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F. This is not a subspace, as it has not the zero vector.

E. This is not a subspace, as it is not closed under addition.

A. This is not a subspace because it doesn\'t contain the zero vector.

B `. This is a subspace. because x = y = 0 gives the zero vector. If c is a scalar, then

c(5x9y, 6x7y, 3x+7y)

= (c(5x9y),c(6x7y),c(3x+7y))

= (5(cx)9(cy), 6(cx)7(cy),3(cx)+7(cy)),

Also,

(5x9y, 6x7y, 3x+7y) + (5a9b, 6a7b,3a+7b)

= (5x9y + 5a9b , 6x7y+6a7b,3x+7y3a+7b)

= (5(x+a)-9(y+b), 6(x+a)-7(y+b), -3(x+a)+7(y+b)),

which is also contained in the set.

D. This is also a subspace. It clearly contains the zero vector. If (x,y,z) and (a,b,c) are in the set, then (x,y,z) + (a,b,c)

= (x+a,y+b,z+c). And

-2(x+a) -8(y+b)

= (-2x-8y) + (-2a-8b)

= 0+0

= 0

we can see that 4(x+a) + 5(z+c) = 0 is similar. So, it is closed under addition.

Finally, let c be a scalar. Then,

c(x,y,z) = (cx,cy,cz), and

-2(cx)-8(cy)

= c(-2x-8y)

= c(0)

= 0

Hence, we can see that 4(cx)+5(cz)=0 is similar.