Subtracting a value from a vector

[Aristarchus_]

Homework Statement

If we have a vector z, what does it mean to subtract a value of 1 from it? What is the geometric interpretation of this?

Relevant Equations

z-1

d

 

[berkeman]

What are your thoughts so far? Do you think it's okay to mix a vector and a scalar in a subtraction operation like that?

 

[Mark44]

Homework Statement:: If we have a vector z, what does it mean to subtract a value of 1 from it? What is the geometric interpretation of this?
Relevant Equations:: z-1

d

It generally makes no sense to add a scalar to a vector or to subtract a scalar from a vector. If z happens to be a complex number, then the expression ##z - 1## is treating 1 as also being a complex number (i.e., 1 + 0i), so both z and 1 are essentially vectors.

 

[fresh_42]

Homework Statement:: If we have a vector z, what does it mean to subtract a value of 1 from it? What is the geometric interpretation of this?
Relevant Equations:: z-1

d

It means that you add two numbers, ##z## and ##1.##

Explanation: You can only add vectors from the same set*). You said that ##z## is a vector. In order to add ##1## to ##z## they have to be of the same dimension*). So ##z+1## demands to be the vector addition in a one-dimensional vector space. Now, a one-dimensional vector space is isomorphic to (~ the same as) the scalar field of the vector space, whatever this is in your case. I assume the real or complex numbers. Therefore ##z+1## is the addition of two numbers, which at the same time are vectors from a one-dimensional vector space.

You see that I already had to make several guesses to answer your question at all! What was missing?

1. Where is ##z## from? Which vector space?
2. What scalar field do you have? Means, if you multiply a vector by a number, where is the number from?
3. What is the dimension of the vector space?
4. Is it a graduated vector space that allows vectors from different dimensions?
5. Is ##1## an abbreviation, e.g. ##1=(1,\ldots,1)##?

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*) This is not 100% true since we can have formal sums like those in a Graßmann algebra, but as you have to ask this question, I allowed myself to make the assumption of a finite-dimensional vector space of one given dimension.