Order, Dimension, Rank, Nullity, Null Space, Column Space of a matrix

hello and welcome to another video we're going to talk about every single concept you see written here we're going to talk about the order Dimension the rank the nullity the column space the null space of a matrix now if you're just starting to learn linear algebra or these terms confuse you and you're not sure where you keep mixing them up this video is meant for you because there was a time I was always confused not just once but for a very long time and then I had to sit down and say what are these things

and what do they mean so I'm going to use this single Matrix to explain all of these terms and I hope it helps you don't forget to like this video share it and be subscribed if you're not leave a comment in the comment section at the end of this video If you gained anything and if you didn't gain anything I need to gain something from your comments still leave the comments in the comment section okay let's get into the video [Music] thank you the first thing we're going to talk about is the order of a

matrix so any Matrix you see has elements in it right like this has nine elements because you can count one two three four five six seven eight nine the number of elements you see in a matrix is the order of the Matrix that's out of the way so we can say number one which is the m by n remember is the number of rows times the number of columns so it is basically row times number of rows times columns we're done let's go to the next thing what is the rank of a matrix because I

don't want to waste time finding the rank because I already did that in another video you want to check this video out I'll put the link in the description because I already explained what the rank of a matrix is I mean how to find the rank of a matrix we're just going to use this because we've already expressed it in row Echelon form and reduce row Echelon form already had videos of these also so what is the rank of this Matrix well the rank of this Matrix is actually the number of linearly independent columns oh

this is important whenever you look at a matrix just look at The Columns of a matrix as guys let's just take them as guys okay now after you put the Matrix in reduce row Echelon form some of those matrices will have one and zeros like this has one and zeros this has one and zeros this one has no one it's all full of zeros I want you to imagine the one with only zeros to be a ghost so every Matrix you see any Matrix you see has real people and ghosts the number of real people

is the rank of The Matrix let that help you every Matrix you see does not mean every column so look at the columns as people standing if they're real at the end they will survive because they're gonna contain ones and zeros but if they have no ones it means the person was not real in the first place it was empty it was made up it was a ghost okay so the number of real people is the rank and that's why we do row Echelon form or reduce row Echelon form now let's compare this to this

has a pivot if you've seen the other video this has a pivot in the first spot so the next one has to be a pivot in the second row the next column and this one has a pivot but if you go here if there's supposed to be anything important it's supposed to be down here but this looks like a repeat of this and you can actually eliminate this row okay so look if you subtract this from this you're going to get this so you see that this really didn't exist on its own and so the

rank of this Matrix is the number of what you call leading columns so a leading column has a leading one okay I explained that when we did the reduce row Echelon form so we're not going to talk all with you the number of pivots that you have this is a pivot this is a pivot and next pivot should be one row down the next column to the right or at least one row down the next column to the right but no there's nothing is the same thing so this is not real okay so that's the

rank of this Matrix let's write the rank so let's talk about the nullity of a Matrix you see the knowledge of a matrix is the number of ghosts in The Matrix I hope that just that that solves the problem the nullity of a matrix is the number of ghosts in a matrix it is the dimension of the set of ghosts now what do I call ghosts the ghost I call are those those columns that end up becoming just zeros okay so look if the rank is to the number of real columns well I shouldn't be

using those expressions but I just want you to understand the concept so tell yourself that so you'll remember that so number four we talk about the nullity the nullity of a is equal to how many do we have here it's one you can also read it from here once something is duplicated and it's not a pivot point here you know that that's going to end up being a zero when you have reduced row Echelon form so you might want to get used to this so what do we have we have the nullity of a is

equal to one and what does it mean it is the number of linearly dependent columns well that will not sound right okay so you can say it is the number of zero columns zero columns in reduced row Echelon form I think that's a better way so it's the number of zero columns you're going to get at the end so let's talk about column space the column space is just it's not here but it's just the the possibilities that you can create with the vectors that you have I call them vectors I'm talking about the columns

okay what can you do the the it is just the space that you can create what linear combinations of these columns can you generate what can you what what else can you make do you think that if I add this to the C if I add this column to this column I'm going to get 3 3 6. three three six that's another column is in the column space what else can I generate well I can put this there I can put this there I can put this there I can subtract this from this I'll get

another Matrix I can subtract or if I subtract this from this I'm definitely going to get this but what if I subtract this two of this from four of this and then add this whatever I can generate by a linear combination of the current columns in this Matrix is what you call a column space so this is usually it can be as as large as possible it doesn't matter what you can do so you call it the span of The Columns of The Matrix so whatever you can generate from a linear combination of all of

the columns in The Matrix you are given any possible linear combination is what you call the column space of the Matrix so let's say number five column space so here what about the null space what is the null space the null space Sounds like this it is the span of all possible things you can generate using the thing that was the null remember the word null is zero or nothing yes or ghost or not real yeah so what can you do with this guy how what what possible linear combinations of this guy can you have

well you can just multiply by a scalar or you leave it there so the span of this Vector that ended up being a ghost is what you call your null space let's put some facts here for everyone to know number one the rank plus the nullity of a matrix the rank of a plus the nullity of a matrix is always equal to the number of columns for a matrix now when we talk about Vector spaces the equivalence is there but I just want you to understand that when you add the rank of a matrix to

the nullity of a matrix you get the number of columns of that Matrix so you always expect that once you get this to be two this must be one and it is clear in a room there are three guys two of them are real the third guy is a ghost now we're going to talk about basis or bases in the next video because that's what I'm going to start we're going to talk about bases and dimensions of vector spaces now there's so many things packed in Vector spaces but it's important that I start talking about

it in the next video now let's see this is number one a very key important thing now look at this expression here the column space when you talk about the span and you're talking about the the linear combination the possible linear combinations of The Columns of a matrix remember that this one actually is a linear combination of these two so this guy is not real that's why you can this this is the x-ray of a matrix and it's just you want to make sure that you make decisions from here the X-ray of this Matrix shows

you that this was real this was real this was a ghost so this was not real so the actually what you were mixing together were the two real columns so what can help you to create your column space is actually not all three of them because this is fake it's these first two so the number of vectors that you need in your column space is what we call the rank and the vectors that you actually need are these two vectors or there could be these two vectors it doesn't matter which Vector you pick because this

can be transformed to look like this okay so it could be this vector or and this Vector so for me I know that this columns this could be written as the span of sorry the span of the set of these vectors was the first Vector one two one and the second one is two one five two one five see the span of this set of vectors is the column space whatever you can do by combining these two will generate any Matrix that looks like this any three by three Matrix like this one thing I have

not talked about it has to do with you looking at a matrix and let's say in what space is the Matrix is it in R1 sorry R1 or is it an R2 or is it in R3 again imagine aliens usually people say oh aliens are very tall they're taller than the average human well that's how I want to imagine the Matrix the taller the Matrix the higher the dimension so this has one two three levels it's in R3 it doesn't matter how many of them are stacked just one of them just count the number of

Parts it has this has three components is an R3 this has if it was a two by two Matrix then it's in R2 if it's a two by five matrix it's still in R2 it just has a bunch of matrices okay so if a matrix is in R3 it will require three matrices I mean three columns that are linearly independent to be able to span R3 we're going to talk about that in another video but basically I think I've covered it so this is an R3 because each column has a height of three three levels

one two three three dimensions it has x y z okay hope this was able to clear some confusion and if you want to know what the span of this the zero columns in reduced row Echelon form would just be this Matrix so if you put this Matrix or this Matrix and you put it here and say it's going to be the span of just one Matrix you'll be correct what can you do with the Matrix whatever you do that's your business I'll see in the next video never stop learning this will stop learning I've stopped

living bye