Dimension

Exercise 1

Suppose \( V \) is finite-dimensional and \( U \) is a subspace of \( V \) such that

\[\dim U = \dim V \]

Prove that \( U = V \)

Exercise 2

Show that the subspaces of \( \mathbb{R}^2 \) are precisely \(\{0\}, \mathbb{R}^2\), and all lines in \( \mathbb{R}^2 \) through the origin.

Exercise 3

Show that the subspaces of \( \mathbb{R}^3 \) are precisely \(\{0\}, \mathbb{R}^3\), all lines in \( \mathbb{R}^3 \) through the origin, and all planes in \( \mathbb{R}^3 \) through the origin.

Exercise 4

  1. Let \( U = \{p \in \mathcal{P}_4(\mathbb{F}) : p(6) = 0\} \). Find a basis of \( U \).
  2. Extend the basis in part (a) to a basis of \( \mathcal{P}_4(\mathbb{F}) \).
  3. Find a subspace \( W \) of \( \mathcal{P}_4(\mathbb{F}) \) such that \( \mathcal{P}_4(\mathbb{F}) = U \oplus W \).

Exercise 5

  1. Let \( U = \{p \in \mathcal{P}_4(\mathbb{R}) : p''(6) = 0\} \). Find a basis of \( U \).
  2. Extend the basis in part (a) to a basis of \( \mathcal{P}_4(\mathbb{R}) \).
  3. Find a subspace \( W \) of \( \mathcal{P}_4(\mathbb{R}) \) such that \( \mathcal{P}_4(\mathbb{R}) = U \oplus W \).

Exercise 6

  1. Let \( U = \{p \in \mathcal{P}_4(\mathbb{F}) : p(2) = p(5)\} \). Find a basis of \( U \).
  2. Extend the basis in part (a) to a basis of \( \mathcal{P}_4(\mathbb{F}) \).
  3. Find a subspace \( W \) of \( \mathcal{P}_4(\mathbb{F}) \) such that \( \mathcal{P}_4(\mathbb{F}) = U \oplus W \).

Exercise 7

  1. Let \( U = \{p \in \mathcal{P}_4(\mathbb{F}) : p(2) = p(5) = p(6)\} \). Find a basis of \( U \).
  2. Extend the basis in part (a) to a basis of \( \mathcal{P}_4(\mathbb{F}) \).
  3. Find a subspace \( W \) of \( \mathcal{P}_4(\mathbb{F}) \) such that \( \mathcal{P}_4(\mathbb{F}) = U \oplus W \).

Exercise 8

  1. Let \( U = \{p \in \mathcal{P}_4(\mathbb{R}) : \int_{-1}^1 p = 0\} \). Find a basis of \( U \).
  2. Extend the basis in part (a) to a basis of \( \mathcal{P}_4(\mathbb{R}) \).
  3. Find a subspace \( W \) of \( \mathcal{P}_4(\mathbb{R}) \) such that \( \mathcal{P}_4(\mathbb{R}) = U \oplus W \).

Exercise 9

Suppose \( v_1, \ldots, v_m \) is linearly independent in \( V \) and \( w \in V \). Prove that

\[\dim \operatorname{span}(v_1 + w, \ldots, v_m + w) \geq m - 1.\]

Exercise 10

Suppose \( p_0, p_1, \ldots, p_m \in \mathcal{P}(\mathbb{F}) \) are such that each \( p_j \) has degree \( j \).
Prove that \( p_0, p_1, \ldots, p_m \) is a basis of \( \mathcal{P}_m(\mathbb{F}) \).

Exercise 11

Suppose that \( U \) and \( W \) are subspaces of \( \mathbb{R}^8 \) such that

\[\dim U = 3, \quad \dim W = 5, \quad \text{and } U + W = \mathbb{R}^8 \]

Prove that \( \mathbb{R}^8 = U \oplus W \).

Exercise 12

Suppose \( U \) and \( W \) are both five-dimensional subspaces of \( \mathbb{R}^9 \). Prove that \( U \cap W \neq \{0\} \).

Exercise 13

Suppose \( U \) and \( W \) are both 4-dimensional subspaces of \( \mathbb{C}^6 \). Prove that there exist two vectors in \( U \cap W \) such that neither of these vectors is a scalar multiple of the other.

Exercise 14

Suppose \( U_1, \ldots, U_m \) are finite-dimensional subspaces of \( V \). Prove that \( U_1 + \cdots + U_m \) is finite-dimensional and

\[ \dim(U_1 + \cdots + U_m) \leq \dim U_1 + \cdots + \dim U_m. \]

Exercise 15

Suppose \( V \) is finite-dimensional, with \(\dim V = n \geq 1\). Prove that there exist 1-dimensional subspaces \( U_1, \ldots, U_n \) of \( V \) such that

\[ V = U_1 \oplus \cdots \oplus U_n. \]

Exercise 16

Suppose \( U_1, \ldots, U_m \) are finite-dimensional subspaces of \( V \) such that \( U_1 + \cdots + U_m \) is a direct sum. Prove that \( U_1 \oplus \cdots \oplus U_m \) is finite-dimensional and

\[ \dim U_1 \oplus \cdots \oplus U_m = \dim U_1 + \cdots + \dim U_m. \]