Linear Algebra (linearly independent)

Question

I want to ask something related to exercise number 5 subchapter 1.5 in linear algebra book by Stephen Friedberg. The question is, "prove that $\{1, x, x^2, \ldots, x^n\}$ is linearly independent in $P_n(F)$"

Is it still true for a finite field? I doubt it.

My counterexample is let's take $F=\mathbb{Z}/2\mathbb{Z}$ as field and evaluate $\{1, x, x^2, \ldots, x^4\}$ In $P_4(F)$. I can take $a_4=a_3=a_2=a_1=1, \text{ and }a_0=0$. I find that $x^4+x^3+x^2+x=0$ for all $x\in F$. Or am I wrong?

Or the question is just simply true because of the implication of zero polynomial resulting all coefficient must be zero? Need help thanks.

The theme recurs. See 1, 2, 3 to get a handle on the meaning of a polynomial. — Jyrki Lahtonen, Aug 28 '19 at 04:38

José Carlos Santos · Accepted Answer · 2019-08-27T07:29:17.387

1

You are wrong, because you are not making a distinction between polynomials and polynomial functions. The simplest example (also over $\mathbb Z_2$) is $x+x^2$. It is a polynomial which is not the null polynomial. However, the polynomial function$$\begin{array}{ccc}\mathbb Z_2&\longrightarrow&\mathbb Z_2\\x&\mapsto&x+x^2\end{array}$$is the null function.

Over infinite fields, this problem doesn't occur.

edited Aug 27 '19 at 07:29

answered Aug 27 '19 at 07:24

José Carlos Santos

427,504

Thank you Jose. So, the proof of the ex. In friedberg satisfy for finite and infinite field? And the proof just like: Since $a_nx^n+\ldots+a_1x+a_0$ equal to zero function for all $a_i\in F$ then $a_nx^n+\ldots+a_1+a_0$ is zero function? Actually im quite confuse. Sorry for asking too much – Ganesha Lapenangga Putra Aug 27 '19 at 11:22
You really should stop thinking about functions. The polynomial $a_0+a_1x+\cdots+a_nx^n$ is the null polynomial if and only if $a_0=a_1=\cdots=a_n=0$. That's all that matters here. – José Carlos Santos Aug 27 '19 at 11:24
Ah ya, i mean zero polynomial. Thank you Jose. Im sorry cant vote for your answer. Im new to this forum – Ganesha Lapenangga Putra Aug 27 '19 at 11:28
I think too hard for this. Thanks by the way – Ganesha Lapenangga Putra Aug 27 '19 at 11:29
Actually i was being affected by this one. https://math.stackexchange.com/questions/1883248/prove-1-x-x2-xn-is-a-linearly-independent-set-in-p-nf – Ganesha Lapenangga Putra Aug 27 '19 at 11:32
You can still mark my answer as the accepted one. Just click on the ✓ sign. – José Carlos Santos Aug 27 '19 at 11:39
Done. Thank you Jose.‍♂️ – Ganesha Lapenangga Putra Aug 27 '19 at 11:49
I am sorry to say this but no, you did not do it. When you do it, the ✓ sign becomes green and your reputations increases. – José Carlos Santos Aug 27 '19 at 11:51
When i mark your answer by clicking the "UP" button, the red light just appear and it said that "thanks for the feedback. Votes cast by those with less than 15 reputation are recorded, but do not change the publicly displayed post score". That said. I just realize that there is a check list button and when i click it, the check list button become green. – Ganesha Lapenangga Putra Aug 27 '19 at 11:56
I hope i can send to you how the red notification appear – Ganesha Lapenangga Putra Aug 27 '19 at 11:58
Thank you. ${}$ – José Carlos Santos Aug 27 '19 at 11:59
I should be the one who said thank you, prof. Jose. – Ganesha Lapenangga Putra Aug 27 '19 at 12:02

score 1 · Answer 2 · answered Aug 27 '19 at 07:31

1

The set of monomials $\{1,x,x^2,\ldots,x^n\}$ is linearly independent over any field $F$.

To see this, one could go back to the definition of polynomials as infinite sequences with almost all (up to finite) entries $0$. If we define $x=(0,1,0,\ldots)$, then with the definition of multiplication of those sequences (polynomials) we obtain $1 =(1,0,0,\ldots)$, $x^2=(0,0,1,0,0,\ldots)$, $x^3=(0,0,0,1,0,0,\ldots)$. From here you see immediately that these monomials are linearly independent over $F$.

answered Aug 27 '19 at 07:31

Wuestenfux

20,964

Thanks Wuestenfux. Sorry dunno how to mention. But may i ask for elementary method please? I mean like is there any method other than using sequences? Will u share with me please? Thanks – Ganesha Lapenangga Putra Aug 27 '19 at 11:26
You can consider polynomials as formal sums which can be manipulated by the usual addition and multiplication. This is what we actually doing and forget the formal definition. – Wuestenfux Aug 27 '19 at 11:29
Thanks wuestenfux – Ganesha Lapenangga Putra Aug 27 '19 at 11:32

Linear Algebra (linearly independent)

2 Answers2