II. Linear Algebra
1. (10%) Suppose $\underline{x}$ and $\underline{y}$ are two linearly independent vectors in $R^n$, and let U be the span of $\underline{x}$ and $\underline{y}$, that is $U = \text{span}\{\underline{x}, \underline{y}\}$. Let $A = \underline{x}\underline{y}^T + \underline{y}\underline{x}^T$.
Prove that the null space N(A) is orthogonal to U, meaning every $\underline{v}$ in N(A) is orthogonal to every $\underline{w}$ in U.