UofM - MATH 2740 - Lecture 11 - Part 2 - Examples of change of basis

UofM - MATH 2740 - Lecture 11 - Part 2 - Examples of change of basis

by Julien A

Formal & Physical Sciences Mathematics PBMathematicsPBFAlgebra
🎙 Julien A 👥 618 📅 April 28, 2022 ⏱ 23 min 👁 659 📄 tutorial 🧭 2026-08-17
Available in: English (current) Français

Keywords

change of basislinear algebracoordinate systemspolynomialsbasis

Summary

This video is a lecture segment from a university linear algebra course, focusing on two examples of change of basis. The instructor begins by introducing two bases for R^2: basis B with vectors u1=(-1,2) and u2=(2,-1), and basis C with vectors v1=(1,0) and v2=(1,1). He then considers a vector x=(5,-1) and shows how to express it in both bases, obtaining coordinates (1,3) in B and (6,-1) in C. He computes the change of basis matrix from B to C by expressing the B basis vectors in terms of the C basis, resulting in columns (-3,2) and (3,-1). He demonstrates that applying this matrix to the coordinate vector in B yields the coordinates in C. The second example involves the vector space P2 of polynomials of degree at most 2. He defines basis B with polynomials 1+x+x^2, x+x^2, and x^2, and basis C as the canonical basis {1, x, x^2}. He expresses a polynomial p(x)=1+x^2 in both bases, obtaining coordinates (1,-1,1) in B and (1,0,1) in C. He then constructs the change of basis matrix from B to C by expressing the B basis polynomials in terms of the canonical basis, yielding an upper triangular matrix. He verifies that multiplying this matrix by the B-coordinate vector gives the C-coordinate vector. The video is a clear, step-by-step tutorial, though the production quality is rough.

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Critical Evaluation

Value of the Information & Strength of the Argument

The video provides valuable worked examples that illustrate the concept of change of basis, which is often abstract. The instructor carefully explains each step, from expressing basis vectors in terms of another basis to constructing the change of basis matrix. The argumentation is solid: he starts with a concrete vector in R^2, then moves to polynomials, showing the generality of the method. The reasoning is clear and logical, with no apparent errors. The examples are well-chosen to demonstrate the process and its application.

Scientific Rigor, Source Quality, Title Accuracy

The video is a lecture, so it does not cite external sources. The mathematical content is standard and correct, aligning with typical linear algebra curricula. The title accurately reflects the content, which is a focused tutorial on change of basis. The production quality is rough (faint pen, audio issues), but this does not affect the mathematical rigor. No comments were provided for analysis.

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Title / Content Match

The title accurately describes the content: two examples of change of basis.

Quality & Reliability

8/10

The video is a clear, step-by-step tutorial on change of basis in linear algebra, with worked examples. The mathematical content is accurate and well-explained, though the production quality is rough (faint pen, audio issues). No external sources are cited, but the content is standard and correct.

Key Moments

Markers derived by PSI from the transcript: the creator did not define chapters.

Contribution & Novelties

The video provides a clear, step-by-step demonstration of change of basis, which is a fundamental concept in linear algebra. It is particularly useful for students who need to see worked examples, as it bridges the gap between abstract theory and practical computation. The second example with polynomials illustrates that the concept applies beyond standard Euclidean spaces, reinforcing the generality of the method.

Pour aller plus loin :

118 words

Radar Profile

The radar profile shows high scores in quality of information and reliability, reflecting the accurate mathematical content. The quantity of information is moderate, as the video is focused on two examples. The technical level is appropriate for an undergraduate linear algebra course.

Reliability 8/10