Vector Spaces

Chickentown

Hi,

Just doing a little bit of vector spaces and I have been following Patrick JMT's videos online in conjunction with the Erwin Kreyszig Advanced Engineering Maths book.

My question is: What is the difference between a row vector and a column vector?

TheBody

Chickentown wrote: »

Hi,

Just doing a little bit of vector spaces and I have been following Patrick JMT's videos online in conjunction with the Erwin Kreyszig Advanced Engineering Maths book.

My question is: What is the difference between a row vector and a column vector?

Not much of a difference really. They both contain the same "information". It just depends on how you want to write a system of equations or how you want to set up a calculation, means you have to choose what way you want to write your vectors.

In particular, for example, say you wanted to multiply a (3 x 3) matrix M, by the vector v=(4,5,6). If we consider v as a row vector, we can only multiply on the left, i.e. (v)(M). However, if we wanted to multiply on the right, we would have to consider v as a column vector in order to do the multiplication, i.e. (m)(v).

You do this kind of thing quite a bit when you are considering things like linear independence, span, etc.

Chickentown

So the column vector V =*[3/4] is equal to the row vector [3 4].(* the '/' is to represent that 3 is sitting on top of 4).

I am stating they are equal because if I plot these two points in the x,y plane I will just in with a 'line' from from (0,0) to (3,4), am I correct?

But we might write it one way or the other depending on the operation you we are trying to do? (Is that what you are saying?)

Thanks for the help, I am trying to get my head around things like span and linear dependence/independence at the moment.

So far in my own words this is what I have:

If we have a set V = { u, w }

And we want to find out if the vectors in the set, namely u and w, are linearly dependent or linearly independent we see if one can be represented as a linear combination of the other. If it can then it is linearly dependent on the other and if it can't then it is linearly independent?

For span I am thinking that if you have some set of vectors in R^2 (*^ indicate superscript 2, not squared) you want to see if the vector you have span the whole plane of R2. Do they cover the whole plane!

As you can see I have a long way to go. Back to the notes.

TheBody

Chickentown wrote: »

So the column vector V =*[3/4] is equal to the row vector [3 4].(* the '/' is to represent that 3 is sitting on top of 4).

I am stating they are equal because if I plot these two points in the x,y plane I will just in with a 'line' from from (0,0) to (3,4), am I correct?

But we might write it one way or the other depending on the operation you we are trying to do? (Is that what you are saying?)

Thanks for the help, I am trying to get my head around things like span and linear dependence/independence at the moment.

So far in my own words this is what I have:

If we have a set V = { u, w }

And we want to find out if the vectors in the set, namely u and w, are linearly dependent or linearly independent we see if one can be represented as a linear combination of the other. If it can then it is linearly dependent on the other and if it can't then it is linearly independent?

For span I am thinking that if you have some set of vectors in R^2 (*^ indicate superscript 2, not squared) you want to see if the vector you have span the whole plane of R2. Do they cover the whole plane!

As you can see I have a long way to go. Back to the notes.

You seem to have the correct idea.

I recommend the book, Elementary Linear Algebra by Howard Anton and Chris Rorres. Very easy to ready and student friendly.

Chickentown

TheBody wrote: »

You seem to have the correct idea.

I recommend the book, Elementary Linear Algebra by Howard Anton and Chris Rorres. Very easy to ready and student friendly.

I'll look into that book. I have an exam in it in 4 weeks. So I am just trying to get the general jist of it now to get through the exams.

I am very interested in it though so I will definitely be looking into it further over the Christmas. Most of the stuff I have been doing up until now has been calculus based and I hadn't given too much thought to sets and the like before.This adds a whole new dimension, literally

TheBody

Chickentown wrote: »

I'll look into that book. I have an exam in it in 4 weeks. So I am just trying to get the general jist of it now to get through the exams.

I am very interested in it though so I will definitely be looking into it further over the Christmas. Most of the stuff I have been doing up until now has been calculus based and I hadn't given too much thought to sets and the like before.This adds a whole new dimension, literally

You seem to be doing ok so far. Be sure to post up any problems you are having and we will help you out if we can.

Good luck!