# Question in regards to Permutation matrices, basis, and non linear functions

• SupaNerd
In summary, to find a basis of R^4 containing the vectors [1 2 3 4] and [0 1 0 1], one can take orthogonal vectors to these two vectors, such as [1 0 0 0] and [0 0 1 0]. These four vectors will be linearly independent and can serve as a basis for R^4. For the second question, a non-linear function f : R^2 -> R^2 can be found by considering two variables for the function, f(x,y), and ensuring that f(rv) = rf(v) for all r and v. Finally, it is clear that applying a permutation matrix on an n vector a = [
SupaNerd
Hey thanks again, figured these questions out!

Last edited:
SupaNerd said:
1) Find a basis of R^4 that contains the vectors [ 1 2 3 4 ] and [ 0 1 0 1].
Note, they are both column vectors but I do not know how to orient them on this site.

let e_1 and e_2 be the 2 vectors you gave.

obviously e_1 and e_2 are lin. indep, so it's possible...just take orthogonal vectors to the previous ones.

But you could just use : e_3=[1 0 0 0] and e_4=[0 0 1 0]...it's obvious that e_1,e_2,e_3 are pairwise orthogonal, hence lin. indep...combining e_2,e_3,e_4 as you want will give you always a vector of the type [c b d b] hence you cannot express e_1 as a combintation of the others, so the e_i are lin. indep...clearly it's a generating subset of R^4...so it's a basis.

2) Find a non linear function f : R^2 ---- > R^2
that still satisfies,
f(rv) = rf(v) for all r and all v.

if f is a function on R^2, then f is a function of 2 variables : f=f(x,y)...you gave only 1 variable for the condition on f.

3) I think it's clear that applying a permutation matrix on an n vector a=[a1...an] corresponds to a permutation of the elements of the vector ?

b=M1*a

Apply a second permutation matrice on b : c=M2*b since c is permutation of b and b a permutation of a, then since the permuation are building a group, c is a permutation of a..hence there exists a permutation matrix M3 such that
c=M3*a...but from above c=M2*M1*a...for all a...then M3=M2*M1...

(This is not a good proof but I cannot think of a brute force computation of direct multipl of two perm. matrices defined as having a 1, and only 1, in every row and column)

Great to hear that you were able to figure out your questions on permutation matrices, basis, and non-linear functions! Permutation matrices are square matrices that represent the rearrangement of rows and columns in a matrix. They are useful in linear algebra for solving equations and finding eigenvalues.

A basis is a set of linearly independent vectors that can be used to express any vector in a vector space. It serves as a starting point for understanding the structure of a vector space and can help with solving problems in linear algebra.

Non-linear functions are functions that do not follow a straight line and cannot be expressed as a linear combination of variables. They are commonly used in fields such as economics, physics, and computer science to model complex relationships between variables.

I hope this helps clarify these concepts for you. Keep up the good work in your studies!

## 1. What is a permutation matrix and how is it used?

A permutation matrix is a square matrix that represents a permutation of a set of numbers. It is used in linear algebra to perform row and column operations on matrices, which can be useful in solving systems of linear equations and finding inverse matrices.

## 2. What is a basis and why is it important in linear algebra?

A basis is a set of linearly independent vectors that can be used to represent and span a vector space. It is important in linear algebra because it allows us to represent and manipulate vectors and matrices in a more efficient and concise way, and it also helps us understand the structure of vector spaces.

## 3. How are permutation matrices related to non-linear functions?

Permutation matrices are not directly related to non-linear functions. However, they can be used in conjunction with other matrices to perform operations on non-linear functions, such as transforming them into linear functions for easier analysis.

## 4. Can permutation matrices be used to solve non-linear equations?

No, permutation matrices are generally used for linear operations and are not suitable for solving non-linear equations. Other techniques, such as substitution or iteration methods, are typically used for solving non-linear equations.

## 5. Are there any limitations to using permutation matrices?

One limitation of permutation matrices is that they can only be used for square matrices, meaning that the number of rows and columns must be equal. Additionally, they can only be used for operations that preserve the number of rows and columns, such as row and column swaps, and cannot be used for operations like scalar multiplication or addition.

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