A system of linear equations in the variables x, y, z has the solution set S = {(x,y, z):x = t, y = 3s, z = 2t with s, te R} a) Is the system non-homogenous or homogenous? Justify your answer b) Determine a basis B for the subspace S of the real vector space R'. Then extend, if necessary, this basis to a basis C for R. Justify why B is a basis for S and C is a basis for R.

A system of linear equations in the variables x, y, z has the solution set S = {(x,y, z):x = t, y = 3s, z = 2t with s, te R} a) Is the system non-homogenous or homogenous? Justify your answer b) Determine a basis B for the subspace S of the real vector space R'. Then extend, if necessary, this basis to a basis C for R. Justify why B is a basis for S and C is a basis for R.

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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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

Vector spaces

A system of linear equations in the variables x, y, z has the solution set
S = {(x,y, z): x = t, y = 3s, z = 2t with s, t e R}
a) Is the system non-homogenous or homogenous? Justify your answer.
b) Determine a basis B for the subspace S of the real vector space R.
Then extend, if necessary, this basis to a basis C for R³.
Justify why B is a basis for S and C is a basis for R3.
c) Is the set C, in 11 b), a basis for S? Justify your answer.

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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