# Optica Hecht Cuarta Edicion Espanol

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Optica Hecht Cuarta Edicion Espanol

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Importance of Vector Space as Quotient of Euclidean Space

This is an excerpt from my textbook about Vector Spaces:

The idea of vector spaces is the idea of an abstract space in which our scalar products is compatible with a quotient operation, $A \oplus B$ where $A$ and $B$ are two different (real or complex) vector spaces. That means that for all points $x \in A, y \in B,$ and a scalar $a$, one has that the “dot product” $a(x+y)$ makes sense.

Given the vector space $A$ and the nonzero vector $x \in A$, the space $A/x$ is indeed a space, in which we can make the quotient operation, and it is a vector space with respect to the usual scalar product.

I am a little confused by the excerpt. Is it telling me that the abstract space, for all points $x \in A$ and a scalar $a$, we have that $a(x+y)$ makes sense? That is, if one identifies the vector space with itself (in the same way we have that the vector space $Z_n$ can be identified with itself in the rational field $\mathbb{Q}$ with the usual scalar product), then for all points $x \in A$ and a scalar $a$, $a(x+y)$ makes sense?

Why do we need to take the quotient of the first vector space to obtain the second vector space?

A:

We choose an additive $+$ and a scalar product on $A$ so that we can make sense of the scalar product $a(x+y)$ where $x$ is some fixed vector of $A$ and $y$ is any vector of $A$.

The vector space $A$ is an abritrary vector space and could be an $n$-dimensional vector space.

The function $a\colon A\to\mathbb{K}$ defined by $a(x+y):=a(x)\cdot a(y)$ where $x$ is a fixed vector and $y$ is a vector in $A$,

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How do we force a bijective function into a set of functions?

For context, I’m just trying to fill the gaps in my knowledge of function theory.

If I have a function $f$

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A:

You could use bash command str to check to see if the file contains a line that begins with abc. In the case where it does, run a similar command to see if the line continues with.jpg.

if [ “abc” -le “$1” ]

then

if [ “$(tail -c 1 ‘ file_content

else

read -p ‘–>’ file_content2

fi

else

echo “Sorry but file does not contain ‘abc'”

fi

See the updated code below for the OP’s updated question. I’ve added the ‘3’ to the grep as I assume you are looking to see if something other than a line starting with abc occurs in the file, and then check if there’s anything after the first line that begins with.jpg.

if [[ “$1” == “ABC” ]]

then

if grep -q.jpg