B Difference between zero and identically zero

[PrathameshR]

What's the difference between something (eg a function or a matrix) becoming zero and it becoming identically zero?
Illustrations will be helpful. Thanks

 

[I like Serena]

Hi PrathameshR ;)

There is no real mathematical distinction.
Identical zero is merely an emphasis to indicate it's 'more' zero than might otherwise be thought.
When we say that a function is identical to zero, we want to emphasize that we really mean the zero-function, which is zero everywhere in its domain.
Saying that a function is zero should mean the same thing (that it's the zero-function), but some authors are a bit sloppy, and they might mean that the function just becomes zero for a certain value in its domain.

Same thing for a matrix.
A matrix that is zero means the same thing as a matrix that is identical to zero - it's the matrix with only zeroes.
That is as opposed to a matrix that has a zero.

 

[PrathameshR]

Hi PrathameshR ;)

There is no real mathematical distinction.
Identical zero is merely an emphasis to indicate it's 'more' zero than might otherwise be thought.
When we say that a function is identical to zero, we want to emphasize that we really mean the zero-function, which is zero everywhere in its domain.
Saying that a function is zero should mean the same thing (that it's the zero-function), but some authors are a bit sloppy, and they might mean that the function just becomes zero for a certain value in its domain.

Same thing for a matrix.
A matrix that is zero means the same thing as a matrix that is identical to zero - it's the matrix with only zeroes.
That is as opposed to a matrix that has a zero.

This really helped. Thanks

 

[Mark44]

There is no real mathematical distinction.
Identical zero is merely an emphasis to indicate it's 'more' zero than might otherwise be thought.

I'm going to disagree slightly with this.
For example the equation $x^2 - 2x + 1 = 0$ is a true statement only when x = 1. OTOH, the equation $x^2 - 2x + 1 - (x - 1)^2 = 0$ is true for all real values of x. The expression $x^2 - 2x + 1 - (x - 1)^2$ is identically zero.

 

[I like Serena]

I'm going to disagree slightly with this.
For example the equation $x^2 - 2x + 1 = 0$ is a true statement only when x = 1. OTOH, the equation $x^2 - 2x + 1 - (x - 1)^2 = 0$ is true for all real values of x. The expression $x^2 - 2x + 1 - (x - 1)^2$ is identically zero.

This is indeed exactly where the ambiguity occurs.
When we write $x^2 - 2x + 1 - (x - 1)^2$, it's somewhat ambiguous if we're talking about a specific function value, or about the function in general.
Literally speaking, the expression $x^2 - 2x + 1 - (x - 1)^2$ is not a function - it's a specific function value.
It's just that it's not uncommon that an author intends the corresponding function.
So saying $x^2 - 2x + 1 - (x - 1)^2$ is identical to zero is intended to mean that $x\mapsto x^2 - 2x + 1 - (x - 1)^2$ is zero.
Or alternatively that the function given by $f(x)=x^2 - 2x + 1 - (x - 1)^2$ is zero.
The word identical here is used to disambiguate, although it's not really what the word identical means (mathematically it just means the same thing as equal to).
Note that 'zero' here is ambiguous as well, since it's not clear whether it's a function value that is zero, or the function itself that is the zero-function.

 

[Mark44]

This is indeed exactly where the ambiguity occurs.
When we write $x^2 - 2x + 1 - (x - 1)^2$, it's somewhat ambiguous if we're talking about a specific function value, or about the function in general.

I didn't say anything about functions, and explicitly wrote "the expression $x^2 - 2x + 1 - (x - 1)^2$.

Literally speaking, the expression $x^2 - 2x + 1 - (x - 1)^2$ is not a function - it's a specific function value.

What I wrote was not in the context of functions, but if you treat it as such, with $f(x) = x^2 - 2x + 1 - (x - 1)^2$, then this is a function that is identically zero. I.e., $f(x) \equiv 0$. That was not my intent, though. Instead, I was distinguishing between an expression that is zero for a particular value of the variable (conditional equality) versus another one that was identically zero.

It's just that it's not uncommon that an author intends the corresponding function.
So saying $x^2 - 2x + 1 - (x - 1)^2$ is identical to zero is intended to mean that $x\mapsto x^2 - 2x + 1 - (x - 1)^2$ is zero.
Or alternatively that the function given by $f(x)=x^2 - 2x + 1 - (x - 1)^2$ is zero.
The word identical here is used to disambiguate, although it's not really what the word identical means (mathematically it just means the same thing as equal to).
Note that 'zero' here is ambiguous as well, since it's not clear whether it's a function value that is zero, or the function itself that is the zero-function.

 

[I like Serena]

I didn't say anything about functions, and explicitly wrote "the expression $x^2 - 2x + 1 - (x - 1)^2$.
What I wrote was not in the context of functions, but if you treat it as such, with $f(x) = x^2 - 2x + 1 - (x - 1)^2$, then this is a function that is identically zero. I.e., $f(x) \equiv 0$. That was not my intent, though. Instead, I was distinguishing between an expression that is zero for a particular value of the variable (conditional equality) versus another one that was identically zero.

Whether we use the word 'function' or not, when we talk about an expression to be identical to zero, we mean the function that assigns a value according to the given expression to each value that is in its domain instead of the actual expression. (And without specification of the domain, it is assumed to be $\mathbb R$ minus any points for which the expression is not defined.)

 

[Mark44]

Whether we use the word 'function' or not, when we talk about an expression to be identical to zero, we mean the function that assigns a value according to the given expression to each value that is in its domain instead of the actual expression.

I don't agree that the concept of functions necessarily needs to be part of such a discussion. One can write $\forall x \in \mathbb R, \sin^2(x) + \cos^2(x) - 1 = 0$ without either implicitly or explicitly stating that the left side is a function. I'm not saying it's wrong to do so, just that it's not necessary.

(And without specification of the domain, it is assumed to be $\mathbb R$ minus any points for which the expression is not defined.)

 

[I like Serena]

I don't agree that the concept of functions necessarily needs to be part of such a discussion. One can write $\forall x \in \mathbb R, \sin^2(x) + \cos^2(x) - 1 = 0$ without either implicitly or explicitly stating that the left side is a function. I'm not saying it's wrong to do so, just that it's not necessary.

When we use the '=' operator, we need to have an equivalence relation that corresponds to it.
And when we add and/or multiply values, we need something like a field (typically $\mathbb R$) in which those are defined.
And when we have an expression with a variable, we need a context to compare it to anything.
That context can either be the field with its associated equality operator (for the 'expression' in your example).
Or it's a function space with its associated equality operator (the 'function' that I refer to).

TL;DR, we leave out that it's a function, but it's a function nonetheless to algebraically treat it as we do.

 

[FactChecker]

"identically zero" means that for all legitimate values of variables, the result is zero. That is quite different from saying that something equals zero for some particular values of variables.

 

[StoneTemplePython]

One more example for OP's original question:

I won't insist on the exclusivity of usage here, just that the common way of using "identically zero" is with respect to polynomials. The idea is that a degree $n -1$ polynomial is completely characterized by $n$ unique data points. (You can prove this using Vandermonde Matrices -- I assume some underlying field with characteristic zero here, probably $\mathbb R$ or $\mathbb C$ for convenience.)

At most $n-1$ of those unique data points may be zeros (aka roots), except in the degenerate case where you're dealing with the zero polynomial (i.e. every thing is identically zero).

i.e. a 'regular' degree $n-1$ polynomial may be written as

$p(x) = c_0 + c_1 x + c_2 x^2 + ... +c_{n-2} x^{n-2}+c_{n-1} x^{n-1}$

if you somehow find that this polynomial has (at least) $n$ unique zeros, that means you in fact have the zero polynomial, i.e.

$p(x) = 0 + 0 x + 0 x^2 + ... +0 x^{n-2}+0 x^{n-1} = 0$
- - - -
I'm not totally comfortable calling the zero matrix "identically zero". I suppose you could argue that the underlying linear transformation has this property. However, note that for an $n$ x $n$ matrix filled entirely with zeros, its characteristic polynomial (like that of all nilpotent matrices) is still

$p(x) = x^{n}$

which is not identically zero.

 

[WWGD]

Maybe a way of making the distinction is whether we have an equation $x^2-2x+1 =0$ or an equality/identity with the same formula. By the Fundamental Thm. of Algebra, above equation ( in one variable) may only have at most 2 zeros.

 

[Mark44]

"identically zero" means that for all legitimate values of variables, the result is zero. That is quite different from saying that something equals zero for some particular values of variables.

I agree completely. There is no need to bring in functions, fields, equivalence operators, or other concepts from advanced algebra.