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