I need to test for a subspace!

How can I determine if the items listed below are subspaces:

1) if B is a subspace of the polynomials where $p(x)$ is an element of the polynomials such that $\int_0^1 xp(x) dx = 0$

2) An $(m\times n)$ matrix A, $\left\{x̄ \in \mathbb{R}^n: Ax̄=0 \right\}$ where 0 is the zero vector in $\mathbb{R}^m$

*For this one I was able to go through the subspace test and prove it is closed under scalar multiplication but I'm not sure how to prove it is closed under addition?

• can you please take a look at the integral in part 1) - I think the "+ 0" is a typo. – Brad S. Mar 21 '14 at 3:55
• Yes it was, I just fixed it, thank you! – Proof Mar 21 '14 at 4:09

2 Answers

You use the subspace test!

Let's do (1) as an example:

Certainly B is nonempty (take the function that is identically zero).

If $p$ and $q$ are polynomials with $\int x p = \int x q = 0$ then $\int x (p + q) = \int x p + \int x q = 0+0 = 0$, and also $p + q$ is a polynomial.

If $p$ is a polynomial and $px$ has zero integral and $\alpha$ is a scalar, then $\int \alpha x p = \alpha \int x p = \alpha \cdot 0 = 0$, and $\alpha p$ is a polynomial.

• I think there should be some $x$'s in the integrals here. – Jason Zimba Mar 21 '14 at 4:40
• Yes thanks. I have edited accordingly. – Frank Mar 21 '14 at 12:36
• Cool - does not change the results, obviously.... – Jason Zimba Mar 21 '14 at 13:06

The kernel of any linear map is a subspace. In your two examples you have the kernels of respectively the linear map $\def\R{\Bbb R}\R[x]\to\R$ given by $p\mapsto\int_0^1 xp(x) dx$ and the linear map $\R^n\to\R^m$ given by $\overline x\mapsto A\overline x$.

Of course this still requires checking that those are indeed linear maps. But that is easy once you've done the check that certain often recurring operations are always linear: matrix multiplication (the second example), mapping polynomials to thier polynomial functions, multiplying functions by a fixed function (here $x$), and integrating functions over a fixed interval (here $[0,1]$). In each case the checks that these are linear maps are entirely straightforward and easy. Note that in the first example you are looking at the composition of three standard linear maps; applying several linear maps in succession always defines a linear map. Thus for most linear maps it is immediate from their definition that they are linear, and as indicated this knowledge also helps recognising many vector subspaces immediately as kernels of linear maps (or as their images, which are also always subspaces).