Understanding Orthonormal Sets and Bases

The video tutorial introduces a set of vectors, focusing on their properties such as normalization and orthogonality. It explains how these vectors form an orthonormal set, which is also linearly independent. The tutorial provides a proof of linear independence and concludes with examples of orthonormal bases using real numbers.

What is the length of each vector in a normalized set?

• 0

• 2

• 3

• 1

What does it mean for vectors to be orthogonal to each other?

• Their dot product is 1

• Their dot product is 0

• They have the same direction

• They have different lengths

What is an orthonormal set?

• A set of vectors with different lengths

• A set of vectors that are both orthogonal and normalized

• A set of vectors that are parallel

• A set of vectors with a length of 2

Why is an orthonormal set linearly independent?

• Because all vectors are parallel

• Because all vectors are orthogonal and non-zero

• Because all vectors have the same length

• Because all vectors are identical

What is the result of the dot product of two different vectors in an orthonormal set?

• 2

• 0

• 1

• 3

What can an orthonormal set be used as in a subspace?

• A vector

• A scalar

• A basis

• A matrix

What is the span of a set of vectors?

• The set of all possible linear combinations of the vectors

• The difference between the vectors

• The sum of the vectors

• The product of the vectors

In the example provided, what is the length of vector v1?

• 0

• 1

• 3

• 2

How do you verify if two vectors are orthogonal?

• Check if their sum is zero

• Check if their dot product is zero

• Check if their difference is zero

• Check if their product is zero

What is the orthonormal basis for a subspace?

• A set of vectors that are neither orthogonal nor normalized

• A set of vectors that are both orthogonal and normalized

• A set of vectors that are orthogonal but not normalized

• A set of vectors that are normalized but not orthogonal