In this video, I review the definition of basis, and discuss the notion of coordinates of a vector relative to that basis.

The properties of a basis of a subspace guarantee that a vector in that subspace can be written as a linear combination of the basis vectors in only one way. The weights in that unique linear combination are called the coordinates of the vector relative to that basis.
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We learned about how vectors can form a basis for a vector space, and we can express any vector within a vector space as a linear combination of the basis vectors. But there can be more than one set of basis vectors. What if we want to express a vector using some other basis rather than the standard basis? Well we can implement a change of basis, so let’s learn how to do that!

Script by Howard Whittle

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(This explanation fits to lectures for students in their first year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)
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When working with vectors we often implicitly use the standard basis. For example, the vector x = [2; -3] tells us the coordinates with respect to the standard basis {[1; 0], [0; 1]}.

At other times it is desirable to work with different coordinate systems that are represented by a different basis. We denote the vector x with respect to the basis B as [x]B.

In this problem we’re provided a basis B and the [x]B. We then compute the vector x, the coordinates of [x]B with respect to the standard basis.

If you enjoyed my videos please “Like”, “Subscribe”, and visit http://adampanagos.org to setup your member account to get access to downloadable slides, Matlab code, an exam archive with solutions, and exclusive members-only videos. Thanks for watching!
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Understanding alternate coordinate systems

Watch the next lesson: https://www.khanacademy.org/math/linear-algebra/alternate_bases/change_of_basis/v/linear-algebra-change-of-basis-matrix?utm_source=YT&utm_medium=Desc&utm_campaign=LinearAlgebra

Missed the previous lesson?
https://www.khanacademy.org/math/linear-algebra/alternate_bases/orthogonal_projections/v/linear-algebra-another-least-squares-example?utm_source=YT&utm_medium=Desc&utm_campaign=LinearAlgebra

Linear Algebra on Khan Academy: Have you ever wondered what the difference is between speed and velocity? Ever try to visualize in four dimensions or six or seven? Linear algebra describes things in two dimensions, but many of the concepts can be extended into three, four or more. Linear algebra implies two dimensional reasoning, however, the concepts covered in linear algebra provide the basis for multi-dimensional representations of mathematical reasoning. Matrices, vectors, vector spaces, transformations, eigenvectors/values all help us to visualize and understand multi dimensional concepts. This is an advanced course normally taken by science or engineering majors after taking at least two semesters of calculus (although calculus really isn’t a prereq) so don’t confuse this with regular high school algebra.

About Khan Academy: Khan Academy offers practice exercises, instructional videos, and a personalized learning dashboard that empower learners to study at their own pace in and outside of the classroom. We tackle math, science, computer programming, history, art history, economics, and more. Our math missions guide learners from kindergarten to calculus using state-of-the-art, adaptive technology that identifies strengths and learning gaps. We’ve also partnered with institutions like NASA, The Museum of Modern Art, The California Academy of Sciences, and MIT to offer specialized content.

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