linear-algebra-kalika102pages-24.pdf

1. Given that $X_1 = \{u_1, u_2, \ldots, u_n\}$ and $X_2 = \{v_1, v_2, \ldots, v_n\}$ are subspaces of a vector space $V$, explain why $\alpha_i u_i \in X_1$ for any scalar $\alpha_i$ and index $i$.

2. Using the expression for $\alpha u + \beta v$ shown in the derivation, identify the coefficient of $u_k$ (for a fixed $k$ between 1 and $n$) in the final linear combination, and write it in terms of $\alpha$, $\beta$, and the original coefficients.

3. Why does the fact that $\alpha_i u_i \in X_1$ and $\beta_i v_i \in X_2$ for all $i$ imply that $\alpha_i u_i + \beta_i v_i \in X_1 + X_2$?

4. Show how the vector $\alpha u + \beta v$ is expressed as a linear combination of the vectors $u_1, \dots, u_n, v_1, \dots, v_n$, using the intermediate steps provided.

5. What property of subspaces ensures that $[X_1 \cup X_2]$ is itself a subspace of $V$, according to the given reasoning?