Let P3 be the vector space defned as polynomials with coeffi

Let P3 be the vector space defned as polynomials with coefficients from R and 3 is the degree of set. Suppose that U1; U2;W are subspaces of P3 such that P3 = U1+W and P3 = U2+W.
Is it true that U1 = U2?

Solution

Given that P3 is a vector space defined as polynomials with coefficients from set of real numbers and 3 is the degree of the set. Say P3 = ax³+bx²+cx+d

Given that U1, U2 and W are subsapces of P3 such that U1+W=P3 and U2+W=P3

Therefore, U1+W=U2+W

Now, we can consider W as any subset of P3. Say for example W = bx²+d

In this case, P3 = U1+W, that is,

ax³+bx²+cx+d = U1 + bx²+d

And from this, we get:

U1 = ax³+cx

Now, likewise, P3 = U2+W, that is:

ax³+bx²+cx+d = U2 + bx²+d

U2 = ax³+cx

Similary, for any subset W of P3, U1 and U2 will be same. Therefore U1=U2