Change of Basis: How to Represent a Vector in a New Set of Basis Vectors | Linear Algebra

What happens when we represent the same vector using a different set of basis vectors? In this video, we visually understand the idea of Change of Basis in Applied Linear Algebra. We start with a vector represented using the standard unit vectors aligned with the x- and y-axes. We then introduce a new set of basis vectors and represent the same vector as a combination of these new vectors. The video shows the change of representation graphically in both coordinate systems: • Representing a vector using the standard basis • Introducing a new set of basis vectors • Expressing the same vector as multiples of the new basis vectors • Visualizing the vector on the new coordinate grid • Seeing how the new basis vectors themselves are represented in the original coordinate system • Understanding why the vector itself does not change — only its coordinates/representation change The key idea is simple: The vector stays the same. The basis changes, so its coordinates change. This visual approach is useful for understanding Change of Basis, Coordinate Systems, Basis Vectors, and Applied Linear Algebra. If you're learning Linear Algebra for engineering, mathematics, computer science, or other applications, this video builds an important foundation for understanding change-of-basis matrices and coordinate transformations. 00:00:00 Introduction Of Problem For Today 00:00:51 Writing the Vector in Standard Way 00:01:50 Defining Other Vectors 00:02:08 Writing In Initial Vectors in terms of Other Vectors 00:03:28 Plotting Initial Vectors in Other Coordinate System 00:04:45 Writing Original Vector in Other Coordinate System 00:06:30 Writing in Initial Coordinate System our Initial Vectors in Terms of Other Vectors #LinearAlgebra #ChangeOfBasis #AppliedLinearAlgebra #Vectors #Basis #Matrix #Mathematics