Let V=R^2 and let H be the subset of V of all points in the first and third quad

Question

Let V=R^2 and let H be the subset of V of all points in the first and third quadrants that lie between the lines y=4xy and y=x/4. Is H a subspace of the vector space V? Let V = R^2 and let H be the subset of V of all points in the first and third quadrants that lie between the lines y = 4x and y = x/4 is H a subspace of the vector space V? Does H contain the zero vector of V? 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 , . 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, . 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.

Explanation / Answer

Let V be vector space over field R

Given that V=R^2 ------------(1)

H be the subset of V of all the points in the first and third quadrant that lie between the lines

y=4x and y=x/4

We will clearly put this in another format that H be the region bounded by inequalities

y<=4x and y>=(x/4)   .. For first quadrant

y>=4x and y<=(x/4)   .. For Third quadrant

Any subset of vector space is called as subspace if it is closed under vector addition and scalar multiplication……………………….(standard def of subspace)

i.e

If a, b belongs to H and X and Y belongs to scalar field say F (here it is R)

Then H is subspace of V if

X.a + Y.b belongs to H

Let (a,b) belongs to H first quadrant.

So it satisfies required constraints.

a/4 <= b<= 4a   …………(2)

Let (c,d) belongs to H as well.

c/4 <= d<= 4c ………….(3)

Adding 2 and 3

(a+c)/4 <= (b+d) <= 4( a+c)

So clearly new point after vector addition (a+c, b+d) satisfies given constraints of H.

Similarly we can show for third quadrant point belongs to H.

So H is closed under vector addition.

Let (a,b) belongs to H first quadrant.

So it satisfies required constraints.

a/4 <= b<= 4a   …………(4)

Let X be scalar belongs to field R of V.

So multiplying eq 4 with scalar X we get,

X * a/4 <= X*b <= X*4a

This inequality does hold true if X is negative and point may belongs to 3rd quadrant.

So H is closed under scalar multiplication.

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