1. For the following problems, check whether the given sets are subspaces. If asked, find a basis of a linear space and thus determine its dimension.
(a) Is the set $\left\{ p(t) : \int_0^1 p(t) dt = 0 \right\}$, a subspace of $P_2$, where $P_2$ denotes the space of all polynomials of degree $\le 2$?
Find a basis for those that are subspaces.
(b) Which of the following subsets V of $\mathbb{R}^{3 \times 3}$ given below are subspaces of $\mathbb{R}^{3 \times 3}$? You don't need to find a basis.
i. The trace of a 3 x 3 matrix A is defined to be the sum of its diagonal entries: i.e., $tr(A) = a_{11} + a_{22} + a_{33}$.
Is the set of all trace zero matrices, i.e., $tr(A) = 0$, a subspace of $\mathbb{R}^{3 \times 3}$?
ii. The 3 x 3 matrices in reduced row-echelon form.
(c) Let V be the space of all infinite sequences of real numbers. Which of the following subsets of V given below are subspaces of V. You don't need to find a basis here.
i. The geometric sequences [i.e., sequences of the form $(a, ar, ar^2, ar^3, \dots)$, for some constants a and r]
ii. The square-summable sequences $(x_0, x_1, \dots)$ (i.e., those for which $\sum_{i=0}^\infty x_i^2$ converges)