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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