## ADDITIONAL PRACTICE PROBLEMS ON 9.1

Additional practice problems on spaces of matrices and polynomials

For each of the following sets of matrices or polynomials, determine whether it is a subspace. If it is, give a basis and determine its dimension. All of these problems are over the REAL numbers.

```                                         1
1. All 3 by 3 matrices   A  such that    1  is in the kernel of  A,
1
1
i.e.  A  1  = 0.
1

```
2. All 4 by 4 matrices such that the first row is 0 and the trace is 0.

3. All 2 by 2 matrices A that commute with the matrix B =

```
0 0
1 0,  i.e.,  such that   AB = BA. ```
4. All 7 by 7 matrices with nonnegative entries.

5. All poylynomials f of degree at most 6 such that f(1) = 0.

6. All polynomials f of degree at most 7 such that f(-x) = -f(x) identically. (E.g., 2x^5 + x^3 + 8x satisfies this condition while x^2 does not.)

7. All polynomials f of degree at most 4 such that f(0) = 0 and f(-1) = 0.

8. All polynomials f of degree 15 such that f(0) = 1.