Differences between row and column vectors

1. Feb 25, 2014

Quarlep

I want to show a vector in matrix but I didnt uderstand differentes between row matrix and column matrix Lets suppose I have a 2i+3j How I will show this vector in matrix ?
I will use a row matrix or column matrix.

2. Feb 25, 2014

Quarlep

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

3. Feb 25, 2014

mathman

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

4. Feb 25, 2014

HallsofIvy

Staff Emeritus
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.

5. Feb 26, 2014

Quarlep

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

6. Feb 26, 2014

Staff: Mentor

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>.
20 vectors in one matrix? Your question is not clear and I have no idea what you're asking.

7. Feb 26, 2014

Staff: Mentor

Maybe like this?
As row vectors: <5, 6> and <3, 9>.
As column vectors:
$\begin{bmatrix} 5 \\ 6\end{bmatrix}$

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

8. Feb 26, 2014

hilbert2

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.

9. Feb 26, 2014

Jhenrique

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!

10. Feb 26, 2014

Quarlep

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.

11. Feb 26, 2014

Quarlep

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

12. Feb 26, 2014

Quarlep

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 ?

13. Feb 26, 2014

Quarlep

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

14. Feb 26, 2014

Jhenrique

15. Feb 26, 2014

Quarlep

I Just didnt understand

16. Feb 26, 2014

Quarlep

I didnt ask you cross product

17. Feb 26, 2014

Staff: Mentor

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.
???

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

18. Feb 26, 2014

Jhenrique

19. Feb 26, 2014

Staff: Mentor

My comment wasn't about a misspelling at all. Below is one example of what I'm talking about.
Your comments aren't always that clear, either, such as this one.
This is pretty inscrutable, IMO.

Last edited by a moderator: Feb 26, 2014
20. Feb 26, 2014

Jhenrique

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.

"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.