Let V be an n-dimensional vector space with basis B = {v1, .•. , vn}. Let P be an invertible n X n matrix and set for i = 1, ... , n. Prove that C = {u1, •.• , un} is a basis for V and show that P = P B-> C

Let V be an n-dimensional vector space with basis B = {v1, .•. , vn}. Let P be an invertible n X n matrix and set for i = 1, ... , n. Prove that C = {u1, •.• , un} is a basis for V and show that P = P B-> C

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter6: Vector Spaces

Section6.3: Change Of Basis

Problem 22EQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Let V be an n-dimensional vector space with basis B = {v₁, .•. , v_n}. Let P be an invertible n X n matrix and set for i = 1, ... , n. Prove that C = {u₁, •.• , u_n} is a basis for V and show that P = P _{B-> C}

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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