- #1
tobor8man
- 2
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Reading in Mathematical Physics by Sadri Hassani. It defines a vector abstractly. I will repeat that definition here rather more informally.
There are these things called vectors, a, b, x etc., that have these properties:
You can add them
a + b = b + a
a + (b + c) = (b + a) + c
a + 0 = a, 0 is the zero vector
a + (- a) = 0
You can multiply them by complex numbers (scalars) like c, d
c(d a) = (cd)a
1 a = a
Multiplication involving vectors and scalars is distributive
c(a + b) = c a+ c b
(c + d) a = c a+ d a
And that is it.
Given that definition, a scalar is a vector, a matrix is a vector, a tensor is a vector. Yes?
Mind you, I have also read that scalars and vectors are a kinds of tensors, of rank 0 and 1 respectively. True?
Am I confused? Should I be?
There are these things called vectors, a, b, x etc., that have these properties:
You can add them
a + b = b + a
a + (b + c) = (b + a) + c
a + 0 = a, 0 is the zero vector
a + (- a) = 0
You can multiply them by complex numbers (scalars) like c, d
c(d a) = (cd)a
1 a = a
Multiplication involving vectors and scalars is distributive
c(a + b) = c a+ c b
(c + d) a = c a+ d a
And that is it.
Given that definition, a scalar is a vector, a matrix is a vector, a tensor is a vector. Yes?
Mind you, I have also read that scalars and vectors are a kinds of tensors, of rank 0 and 1 respectively. True?
Am I confused? Should I be?