Problem Suppose $U_1, \dots, U_m$ are finite-dimensional subspaces of $V$ such that $U_1 + \dots + U_m$ is a direct sum. Prove that $U_1 \oplus \dots \oplus U_m$ is finite-dimensional and $$\dim (U_1 \oplus \dots \oplus U_m) = \dim U_1 + \dots + \dim U_m$$ Solution Proceed by induction on $m$ BC: for $m = 1$ we have $\dim U_1 = \dim U_1$ IH: suppose that it holds for some $m = k \in \mathbb{N}$ that $\dim (U_1 \oplus \dots \oplus U_k) = \dim U_1 + \dots + \dim U_k$...
Problem Suppose $V$ is finite-dimensional, with $\dim V = n \geq 1$. Prove that there exist 1-dimensional subspaces $U_1, \dots, U_n$ of $V$ such that $$V = U_1 \oplus \dots \oplus U_n$$ Solution Let $u_1, \dots, u_n$ be a basis of $V$, then $U_i = \textrm{span}(u_i)$ has the property that $\sum_{i=1}^{j}U_i \cap U_{j+1} = \lbrace 0 \rbrace$ for all $j \in \lbrace 1,2, \dots, n-1 \rbrace$ by the linear independence of $u_1, \dots, u_n$ and also $U_1 + \dots + U_n \stackrel{D1....
Problem Suppose $U_1, \dots, U_m$ are finite-dimensional subspaces $V$. Prove that $U_1 + \dots + U_m$ is finite-dimensional and $$\dim (U_1 + \dots + U_m) \leq \dim U_1 + \dots + \dim U_m$$ Solution Theorem 1.39 implies that $U_1 + \dots + U_m$ is a subspace of $V$ and theorem 2.26 therefore ensure that $U_1 + \dots + U_m$ is finite dimensional. To prove the second part: $$\dim U_1 + \dots + U_{m-1} + U_m \stackrel{T2....
Problem Suppose $U$ and $W$ are both 4-dimensional subspaces of $\mathbb{C}^6$. Prove that there exist two vectors in $U \cap W$ such that neither of these vectors is a scalar multiple of the other Solution By theorem 2.43, 2.38 and the fact that $U + W$ is a subspace of $\mathbb{C}^6$ by theorem 1.39: $$6 = \dim \mathbb{C}^6 \geq \dim (U + W) = \dim U + \dim W - \dim U \cap W$$ $$= 4 + 4 - \dim U \cap W$$ Implying that $\dim U \cap W \geq 2$, therefore by the definition of dimension....
Problem Suppose $U$ and $W$ are both five-dimensional subspaces of $\mathbb{R}^9$. Prove that $U \cap W \neq \lbrace 0 \rbrace$ Solution By theorem 2.43, 2.38 and the fact that $U + W$ is a subspace of $\mathbb{R}^9$ by theorem 1.39: $$9 = \dim \mathbb{R}^9 \geq \dim (U + W) = \dim U + \dim W - \dim U \cap W$$ $$= 5 + 5 - \dim U \cap W$$ Implying that $\dim U \cap W \geq 1$....