Math Challenges

Submissions for Problem #19

Problem #19

\[ \displaylines{\text{If }k\ge2\text{ and }\mathbf{x}\in\mathbb{R}^{k}\text{ show that there exists }\mathbf{y}\in\mathbb{R}^{k}\text{ such that }\mathbf{y}\ne\mathbf{0}\text{ but }\mathbf{x}\cdot\mathbf{y}=0\\ \text{Is this also true if }k=1?\\ } \]

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zapwai

Solution: \( \displaylines{\text{Suppose }x_{i}\text{ and }x_{j}\text{ are nonzero (and possibly others).}\\ \text{ Let }y_{i}=-x_{j}\text{ and }y_{j}=x_{i}\text{ and all other y coordinates equal zero.}\\ \text{Then }\mathbf{x}\cdot\mathbf{y}=-x_{j}x_{i}+x_{i}x_{j}=0.} \)

Explanation:
If x is nonzero in only one coordinate, let the corresponding y coordinate equal zero, and some other y coordinate be nonzero. This does not work in one dimension, as there are no other coordinates to let be nonzero for y.