# Which of the following sets are subspaces of R3?

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{(-9x -2y, 3x – 4y, -6x + 4y) I x,y arbitrary numbers}

{(x,y,z) I -3x + 6y + 2z + 9}

{(x,y,z) I -7x + 5y = 0, -8x – 9z = 0}

{(x,y,z) I 7x – 5y + 8z = 0}

{(x, x + 7, x – 5) I x arbitrary number}

{(x,y,z) I x < y < z}

If you can give an explanation too because I don’t quite get this

• I will number these six sets from i) to vi).

i) (-9x -2y, 3x – 4y, -6x + 4y) = x(-9, 3, -6) + y(-2, -4, 4), and x and y are arbitrary.

So this set is the span of the vectors (-9, 3, -6) and (-2, -4, 4). The span of any collection of vectors is always a subspace, so this set is a subspace.

ii) This problem doesn’t make sense, since -3x + 6y + 2z + 9 has no equals sign and therefore is not a condition. If this problem were modified correctly, I would suspect that the set is not a subspace because of the nonzero constant term 9.

iii) and iv) are solution sets of systems of linear equations with zeros for all the right-hand constants and therefore must be subspaces, since the solution set of any system of linear equations with zeros for all the right-hand constants is always a subspace.

v) This set is not a subspace, since it is not closed under scalar multiplication.

For example, (0, 7, -5) is in the set, but 2(0, 7, -5) = (0, 14, -10) is not in the set.

vi) This set is not a subspace, since it is not closed under scalar multiplcation.

For example, (2, 3, 4) is in the set, but (-1)(2, 3, 4) = (-2, -3, -4) is not in the set.

Lord bless you today!

RE:

Which of the following sets are subspaces of R3?

{(-9x -2y, 3x – 4y, -6x + 4y) I x,y arbitrary numbers}

{(x,y,z) I -3x + 6y + 2z + 9}

{(x,y,z) I -7x + 5y = 0, -8x – 9z = 0}

{(x,y,z) I 7x – 5y + 8z = 0}

{(x, x + 7, x – 5) I x arbitrary number}

{(x,y,z) I x < y < z}

If you can give an explanation too because I don’t quite get this

• sets subspaces r3

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