# Differences between row and column vectors

• Quarlep
In summary, to write a vector in terms of a matrix, you would need to use a transformation that transforms a row matrix into a column matrix or a column matrix into a row matrix.
Quarlep
I want to show a vector in matrix but I didnt uderstand differentes between row matrix and column matrix Let's suppose I have a 2i+3j How I will show this vector in matrix ?
I will use a row matrix or column matrix.

And I want to add something If I have two vectors 5i+6j and 3i+9j how I will show them ?

It depends on what you want to do with the matrix. By itself a vector can be expressed either way.

As mathman said, a "row" and "column" vectors are just ways to "express" or "represent" vectors. Which you use depends upon you and what you want to do with them.

One common application is this: Given any vector space, V, of dimension n, the set of all linear functions that map vectors to real numbers is also a vector space of dimension n, called the "dual space" to V. We can show that by writing the vectors in V as "column vectors" and functions in the dual space as "row vectors" so that if the vector is $\begin{bmatrix}x \\ y \\ z\end{bmatrix}$ and the function is $\begin{bmatrix}a & b & c\end{bmatrix}$ then the action of the function on the vector is the matrix product
$$\begin{bmatrix}a & b & c\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}= ax+ by+ cz$$

But, again, that is simply a convenient way of representing vectors.

I want to show 20 vectors in ne matrix How can I really do that

Quarlep said:
I want to show a vector in matrix but I didnt uderstand differentes between row matrix and column matrix Let's suppose I have a 2i+3j How I will show this vector in matrix ?
I will use a row matrix or column matrix.
A matrix is a rectangular block of entries of some kind, with rows going across, and columns going vertically. You can have row vectors or column vectors, but not row matrices or column matrices.

You can write 2i + 3j as <2, 3>.
Quarlep said:
I want to show 20 vectors in ne matrix How can I really do that
20 vectors in one matrix? Your question is not clear and I have no idea what you're asking.

Quarlep said:
And I want to add something If I have two vectors 5i+6j and 3i+9j how I will show them ?
Maybe like this?
As row vectors: <5, 6> and <3, 9>.
As column vectors:
## \begin{bmatrix} 5 \\ 6\end{bmatrix}##

## \begin{bmatrix} 3 \\ 9\end{bmatrix}##

I want to show a vector in matrix but I didnt uderstand differentes between row matrix and column matrix

Well, for example, if ##x## is a row vector and ##y## is a column vector, the the product ##xy## is an inner product and is a number, while the product ##yx## is an outer product and is a matrix.

Answering partially you ask. colum vector represents vector quantities of kind "meters for ...", where the unid of length appears in the numerator. row vector represents vector quantities of kind "... for meters", where the unid of length appears in the denominator.

So, the correct representation for ##\vec{\nabla}## isn't ##\begin{bmatrix}
\frac{\partial }{\partial x}\\
\frac{\partial }{\partial y}\\
\end{bmatrix}## and yes: ##\begin{bmatrix}
\frac{\partial }{\partial x} & \frac{\partial }{\partial y}
\end{bmatrix}##

Say that a vector haven't multiplicative inverse, but for effect of calculus, this affirmation is sux, because exist inverse of vector yes (https://en.wikipedia.org/wiki/Curvilinear_coordinates#Covariant_and_contravariant_bases). The inverse of unit vector ##\hat{q}_i## is ##\frac{1}{\hat{q}_i} =\hat{q}^i##. To invert a vector needs too to invert your matrix representation, i. e., get the transpose representation.

Vector/matrix calculus/algebra is a disorder!

I didnt get my answer beacuse I couldn't tell you my ideas exactly.I want to tell you I have a vector a "normal"vector not a row vector or not a column vector.I have just normal vector.Then I want to show this normal vector in Matrix system.

And my previous idea is about this I want to show some vectors in Matrix

And I have another question I have two Matix one of them size is 1x6 and other one is 2x1 so How can I transform this two Matrix into vector.Because If I try 1x6 Matrix transorm into vector I have to use 6 dimension or not ?

Actually you are help me but I am not sure which one is true or trustable

I Just didnt understand

I didnt ask you cross product

Quarlep said:
And I have another question I have two Matix one of them size is 1x6 and other one is 2x1 so How can I transform this two Matrix into vector.
They already are vectors. A matrix with only one row or one column is usually called a vector. A 1 X 6 matrix is a row vector. A 2 X 1 matrix is a column vector.
Quarlep said:
Because If I try 1x6 Matrix transorm into vector I have to use 6 dimension or not ?
?

Probably difficulty with English, but you're not making a whole lot of sense.

Quarlep said:
I didnt ask you cross product

I haven't guilt that the answer is in the session of cross product. See this link with more attention: https://upload.wikimedia.org/math/3/8/e/38e1feb99d143b91383d2fb96ce8e10f.png

Mark44 said:
Probably difficulty with English

Just because the guy ate one f in "transorm". I think this kind of comment is irrelevant and common in this forum.

Just because the guy ate one f in "transorm". I think this kind of comment is irrelevant and common in this forum.
My comment wasn't about a misspelling at all. Below is one example of what I'm talking about.
Quarlep said:
How can I transform this two Matrix into vector.Because If I try 1x6 Matrix transorm into vector I have to use 6 dimension or not ?
Your comments aren't always that clear, either, such as this one.
Jhenrique said:
I haven't guilt that the answer is in the session of cross product.
This is pretty inscrutable, IMO.

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humm... indeed, the english this guy isn't very compatible with the grammar... kkkkkk

But, how english isn't my natural idiom I didn't notice the fails. Actually, I remember that I read in somewhere that when you commits some simple wrong, like exchange one letter (not the first and the last) of a word in middle of a text, your mind reads such word like if the wrong no exist.

Mark44 said:
Your comments aren't always that clear, either, such as this one.
This is pretty inscrutable, IMO.

"I haven't guilt" is a expression/justification for the "answer" (my answer in post #14 that is the answer for his ask too) is in the session (of wiki) of cross product.

My answer is full of reference. Maybe really hard of interpret. I don't know.

Quarlep said:
I want to show a vector in matrix but I didnt uderstand differentes between row matrix and column matrix Let's suppose I have a 2i+3j How I will show this vector in matrix ?
I will use a row matrix or column matrix.
The set of 2x1 matrices and the set of 1x2 matrices is the same set. It's just ##\mathbb R^2##. The row notation and the column notation for an element of ##\mathbb R^2## are just ways to indicate what multiplication operations you intend to apply to it.

If you denote your basis vectors by i,j, and write an arbitrary vector v as ##v=ai+bj##, then the numbers a,b are called the components of v with respect to the ordered basis (i,j). The matrix of components of v with respect to (i,j) is, by convention, the column matrix ##\begin{pmatrix}a\\ b\end{pmatrix}##.

You can use a row matrix instead of a column matrix if you want to. There is however a reason to prefer the column matrix:

Linear transformations can also be represented by matrices. (See the https://www.physicsforums.com/showthread.php?t=694922 about this). Suppose that L is a linear transformation and that we want to represent v, L and Lv by matrices. I will denote them by [v], [L] and [Lv]. If we take [v] and [Lv] to be column matrices, we can define [L] so that [Lv]=[L][v]. But if we take [v] and [Lv] to be row matrices, the best we can do is to define [L] so that we get [Lv]=[v][L].

Quarlep said:
And I want to add something If I have two vectors 5i+6j and 3i+9j how I will show them ?
Are you saying that you want to write (5,6)+(3,9)=(8,15) as a matrix equality? It doesn't matter if you choose to write them as rows or as columns, as long as you make the same choice for all of them. Either write all three as rows, or write all three as columns.

Quarlep said:
And I have another question I have two Matix one of them size is 1x6 and other one is 2x1 so How can I transform this two Matrix into vector.Because If I try 1x6 Matrix transorm into vector I have to use 6 dimension or not ?
It sounds like you want to transform two vectors from two different vector spaces into a single vector. There's no natural way to do that. What I mean by that is that you can certainly let u and v be vectors from two different spaces, and f a function such that f(u,v) is a vector in a third space, but there's no preferred function f that you should use to do this.

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Fredrik said:
The set of 2x1 matrices and the set of 1x2 matrices is the same set. It's just ##\mathbb R^2##. The row notation and the column notation for an element of ##\mathbb R^2## are just ways to indicate what multiplication operations you intend to apply to it.

If you denote your basis vectors by i,j, and write an arbitrary vector v as ##v=ai+bj##, then the numbers a,b are called the components of v with respect to the ordered basis (i,j). The matrix of components of v with respect to (i,j) is, by convention, the column matrix ##\begin{pmatrix}a\\ b\end{pmatrix}##.

You can use a row matrix instead of a column matrix if you want to. There is however a reason to prefer the column matrix:

Linear transformations can also be represented by matrices. (See the https://www.physicsforums.com/showthread.php?t=694922 about this). Suppose that L is a linear transformation and that we want to represent v, L and Lv by matrices. I will denote them by [v], [L] and [Lv]. If we take [v] and [Lv] to be column matrices, we can define [L] so that [Lv]=[L][v]. But if we take [v] and [Lv] to be row matrices, the best we can do is to define [L] so that we get [Lv]=[v][L].

Are you saying that you want to write (5,6)+(3,9)=(8,15) as a matrix equality? It doesn't matter if you choose to write them as rows or as columns, as long as you make the same choice for all of them. Either write all three as rows, or write all three as columns.

It sounds like you want to transform two vectors from two different vector spaces into a single vector. There's no natural way to do that. What I mean by that is that you can certainly let u and v be vectors from two different spaces, and f a function such that f(u,v) is a vector in a third space, but there's no preferred function f that you should use to do this.

You really help me Thanks .I want to ask you something I have a 1x6 matrix and I want to transform it a vector I will wrote it like this (a,b,c,d,e,f)

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Quarlep said:
You really help me Thanks .I want to ask you something I have a 1x6 matrix and I want to transform it a vector I will wrote it like this (a,b,c,d,e,f)
Or you could write it like this: <a, b, c, d, e, f>.

Again, a 1 X 6 matrix is a vector.

whats the difference between write like this <ab,c,d,e,f> and this (a,b,c,d,e,f)

Not much, as long as it is understood that you're talking about vectors. I use angle brackets (< >) so that something like <1, 3, 2> looks different from (1, 3, 2), which could be misinterpreted as a point in R3.

Some people use the notation (x,y) for ordered pairs. Some people don't like that, because (x,y) is also used to denote an open interval, the set of real numbers r such that x < r < y. So they use <x,y> instead (and <x,y,z> for ordered triples). I prefer not to do that, because <x,y> is also used to denote the inner product of two vectors x and y.

These are just notational conventions, so use whatever you want.

## 1. What is the difference between a row vector and a column vector?

A row vector is a one-dimensional array of numbers arranged horizontally, while a column vector is a one-dimensional array of numbers arranged vertically.

## 2. Can a row vector and a column vector have the same number of elements?

Yes, both row and column vectors can have the same number of elements. The main difference is in their orientation, not in the number of elements.

## 3. How are row and column vectors used in linear algebra?

In linear algebra, row vectors are often used to represent a set of equations or constraints, while column vectors are used to represent variables or unknowns in those equations.

## 4. Can a row vector be converted into a column vector, and vice versa?

Yes, a row vector can be converted into a column vector by transposing it, which means switching its rows and columns. Similarly, a column vector can be converted into a row vector by transposing it.

## 5. When should row or column vectors be used in scientific research?

Row and column vectors are commonly used in scientific research when working with matrices, as they allow for easy manipulation and calculation of data. They can also be used to represent data in a compact and organized manner.

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