To find a basis for W, we need to determine the linearly independent polynomials that generate the space.
First, let's rewrite the given polynomials:
P1(x) = x^3 + 3x^2 - x + 4
P2(x) = 2x^3 + x^2 - 7x - 7
We can see that both polynomials have degree 3, so the basis of W must include at least one polynomial of degree 3. Let's start by considering P1(x):
P1(x) = x^3 + 3x^2 - x + 4 = 1*x^3 + 3*x^2 - 1*x + 4
We can write P1(x) as a linear combination of the powers of x:
P1(x) = 1*x^3 + 3*x^2 - 1*x + 4 = 1*x^3 + 3*x^2 + (-1)*x + 4*x^0
So, P1(x) can be generated by the monomials {x^3, x^2, x^0}. These monomials are linearly independent since none can be expressed as a combination of the others.
Now let's consider P2(x):
P2(x) = 2*x^3 + x^2 - 7*x - 7 = 2*x^3 + 1*x^2 + (-7)*x + (-7)*x^0
So, P2(x) can be generated by the monomials {x^3, x^2, x^0}. These monomials are also linearly independent.
Since both polynomials can be generated by the same set of linearly independent monomials, we can conclude that any other polynomial in the space W can also be generated by these monomials. Therefore, a basis for W is {x^3, x^2, x^0}.
The dimension of a vector space is equal to the number of elements in its basis. In this case, the dimension of W is 3, as the basis {x^3, x^2, x^0} has 3 elements.
Let W be the space generated by the polynomials x3 + 3x2 – x + 4, and
2x3 + x2 – 7x – 7. Find a basis and the dimension of W.
1 answer