2. If the equation Gx = y has more than one solution for some y & R", can the columnsof G span R"? (you can use Invertible Matrix Theorem)Note. If the equation Hx = c is inconsistent for some c = R", what can you say aboutthe equation? (you do not need to submit).

Sol:- To understand why the columns of G cannot span Rn if Gx = y has more than one solution for…

Answered: 2. If the equation Gx = y has more than…

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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Related questions

Question

2. If the equation Gx = y has more than one solution for some y & R", can the columns
of G span R"? (you can use Invertible Matrix Theorem)
Note. If the equation Hx = c is inconsistent for some c = R", what can you say about
the equation? (you do not need to submit).

Expert Solution

Step 1

Sol:-

To understand why the columns of G cannot span Rⁿ if Gx = y has more than one solution for some y ∈ Rⁿ, we need to use the Invertible Matrix Theorem. The Invertible Matrix Theorem states that a square matrix G is invertible if and only if its columns form a linearly independent set and span Rⁿ. If G is invertible, then for every vector y ∈ Rⁿ, the equation Gx = y has a unique solution given by x = G^-1y, where G^-1 is the inverse of G.