In Exercises 3–12, determine whether each set equipped

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2022-02-06

In Exercises 3–12, determine whether each set equipped with the given operations is a vector space. For those that are not vector spaces identify the vector space axioms that fail. The set of all pairs of real numbers of the form (x, y), where , with the standard operations on .

Answer & Explanation

nick1337

nick1337

Expert2022-03-08Added 777 answers

We know that with standard operations on R2R2 is a vector space.

Let S={(x,0) | xR}

Note that SR

If we show S is closed under addition & scalar multiplication, Then S will be a vector space over R.

For (x,0) & (y,0)S

(x,0)+(y,0)=(x+y,0)S

Since (x,0),(y,0)S for all (x,0),(y,0)S.

Now choose RR& (x,0)S

Then k(x,0)=(kx,0)S

Since kR& (x,0)S were arbitrary

k(x,0)S for all kR&(k,0)S

S is closed under addition & scalar multiplication

S is a vector space over R or you can S is a vector subspace of R2

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