Let V = R^2 and let H be the subset of V of all points on the line -3x + 2y = -6

Question

Let V = R^2 and let H be the subset of V of all points on the line -3x + 2y = -6. Is H a subspace of the vector space V? 1. Is H nonempty? H is empty 2. Is H closed under addition? If it is, enter CLOSED. If it is not, enter two vectors in H whose sum is not in H, using a comma separated list and syntax such as , . 3. Is H closed under scalar multiplication? If it is, enter CLOSED. If it is not, enter a scalar in R and a vector in H whose product is not in H, using a comma separated list and syntax such as 2, . 4. Is H a subspace of the vector space V? You should be able to justify your answer by writing a complete, coherent, and detailed proof based on your answers to parts 1-3. H is a subspace of V

Explanation / Answer

1) H is empty

2) -3x + 2y = -6 ==> y = (-6 + 3x)/2

Lets take two points u(2, 0), v( 4, 3)

Now add these two we get the point (6, 3) which does not satisfy the equation

So H is not closed under addition

3) Now take point u(2,0)

So r * u ==> (2r, 0 ) take r = 2 ==> (4,0) which doesn't satisfy the equation

So H is not closed under scalar multiplication

4) H is not a subspace of vector V as it does not satisfy addition and multiplication

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Thank you

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