Data Science and Machine Learning (Theory and Projects) A to Z - Mathematical Foundation: Orthonormal Basis

The video tutorial explains the concept of vector spaces and the importance of orthonormal bases. It covers orthogonal vectors, dot products, and the process of normalization. The tutorial also introduces the concept of orthonormal bases, highlighting their computational efficiency. Finally, it discusses the Gram-Schmidt orthogonalization process for converting a basis to an orthonormal basis.

Why is it beneficial to use an orthonormal basis in vector spaces?

• It simplifies the visual representation of vectors.

• It allows for more efficient computations.

• It reduces the number of vectors needed.

• It increases the dimensionality of the space.

What is the condition for two vectors to be orthogonal?

• Their cross product is zero.

• Their magnitudes are equal.

• Their dot product is zero.

• They lie on the same line.

How can you make two vectors orthogonal?

• By adding a constant to both vectors.

• By changing one entry in one of the vectors.

• By multiplying both vectors by a scalar.

• By rotating one of the vectors by 90 degrees.

What is the norm of a vector?

• The difference between its maximum and minimum components.

• The sum of its components.

• The square root of the sum of the squares of its components.

• The product of its components.

What does it mean for a vector to be normalized?

• It has the same direction as the original vector.

• It is orthogonal to all other vectors.

• Its norm is equal to one.

• Its components are all integers.

What are the conditions for a set of vectors to form an orthonormal basis?

• All vectors have the same magnitude and direction.

• All vectors are orthogonal and have a norm of one.

• All vectors are parallel and have different magnitudes.

• All vectors are perpendicular and have a norm greater than one.

What is the Gram-Schmidt process used for?

• To find the inverse of a matrix.

• To convert a set of vectors into an orthonormal basis.

• To calculate the determinant of a matrix.

• To solve systems of linear equations.