## Linear Algebra and Its Applications, Exercise 3.1.2

Exercise 3.1.2. For $\mathbb{R}^2$ give an example of linearly independent vectors that are not mutually orthogonal, as well as mutually orthogonal vectors that are not linearly independent.

Answer: The vectors $(1, 0)$ and $(1, 1)$ are linearly independent, since the second vector cannot be expressed as a scalar times the first vector. However the two vectors are not orthogonal since their inner product is $1 \cdot 1 + 1 \cdot 0 = 1$.

The zero vector $(0, 0)$ is orthogonal to every vector in $\mathbb{R}^2$, including itself, but is not linearly independent of such vectors.

NOTE: This continues a series of posts containing worked out exercises from the (out of print) book Linear Algebra and Its Applications, Third Edition by Gilbert Strang.

If you find these posts useful I encourage you to also check out the more current Linear Algebra and Its Applications, Fourth Edition , Dr Strang’s introductory textbook Introduction to Linear Algebra, Fourth Edition and the accompanying free online course, and Dr Strang’s other books .

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