0.1 Practice Problems

1.

Consider the set \(S = \{ (x, y, z) \in \mathbb {R}^3 \mid x = 0 \}\). Identify whether the set is a vector space under the standard vector addition and scalar multiplication.

2.

Determine whether \(S = \{ f \in \mathbb {R} \mid f'(0) = 0 \}\) is a vector space for differentiable functions \(f\).

3.

For all \(n \in \mathbb {Z}\) and \(a_i \in \mathbb {R}\), determine whether \[ S = \left \{ (x_1, x_2, \ldots , x_n) \in \mathbb {R}^n \,\middle \vert \, \sum _{k = 1}^n a_k x_k = 0 \right \} \] is a vector space.

4.

Show that the intersection of subspaces of a vector space \(V\) is also a subspace of \(V\).

5.

Show that for distinct real numbers \(a_1, \ldots , a_n\), the set \(\{ e^{a_1 x}, \ldots , e^{a_n x} \}\) is linearly independent.

6.

Let \(A, B \in \mathbb {R}^3\) be lines defined as \(x = y = z\) and \(3x = 4y = 5z\) respectively. If \(C = \{ a + b \mid a \in A, b \in B \}\), show that \(C = A \oplus B\) where \(\oplus \) represents the direct sum [MATH02-3].

7.

If \(\{ a, b, c \}\) is linearly independent in a vector space \(V\), show that \(\{ a+b, b+c, c+a \}\) is also linearly independent [MATH02-3].

8.

Show that the set of all polynomials in \(\mathbb {P}^3\) that includes the origin is a subspace of \(\mathbb {P}^3\).