Column Space of a Matrix Introduction to the column space of a matrix
Column Space of a Matrix
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- We spent a good deal of time on the idea of a null space.
- What I'm going to do in this video is introduce you to a
- new type of space that can be defined around a matrix, it's
- called a column space.
- And you could probably guess what it means just based on
- what it's called.
- But let's say I have some matrix A.
- Let's say it's an m by n matrix.
- So I can write my matrix A and we've seen this multiple
- times, I can write it as a collection of columns vectors.
- So this first one, second one, and I'll have n of them.
- How do I know that I have n of them?
- Because I have n columns.
- And each of these column vectors, we're going to have
- how many components?
- So v1, v2, all the way to vn.
- This matrix has m rows.
- So each of these guys are going to have m components.
- So they're all members of Rm.
- So the column space is defined as all of the possible linear
- combinations of these columns vectors.
- So the column space of A, this is my matrix A, the column
- space of that is all the linear combinations of these
- column vectors.
- What's all of the linear
- combinations of a set of vectors?
- It's the span of those vectors.
- So it's the span of vector 1, vector 2, all the
- way to vector n.
- And we've done it before when we first talked
- about span and subspaces.
- But it's pretty easy to show that the span of any set of
- vectors is a legitimate subspace.
- It definitely contains the 0 vector.
- If you multiply all of these guys by 0, which is a valid
- linear combination added up, you'll see that it contains
- the 0 vector.
- If, let's say that I have some vector a that is a member of
- the column space of a.
- That means it can be represented as some linear
- So a is equal to c1 times vector 1, plus c2 times vector
- 2, all the way to Cn times vector n.
- Now, the question is, is this closed under multiplication?
- If I multiply a times some new-- let me say I multiply it
- times some scale or s, I'm just picking a random letter--
- so s times a, is this in my span?
- Well s times a would be equal to s c1 v1 plus s c2 v2, all
- the way to s Cn Vn Which is once again just a linear
- combination of these column vectors.
- So this Sa, would clearly be a member of the
- column space of a.
- And then finally, to make sure it's a valid subspace-- and
- this actually doesn't apply just to column space, so this
- applies to any span.
- This is actually a review of what we've done the past. We
- just have to make sure it's closed under addition.
- So let's say a is a member of our column space.
- Let's say b is also a member of our column space, or our
- span of all these column vectors.
- Then b could be written as b1 times v1, plus b2 times v2,
- all the way to Bn times Vn.
- And my question is, is a plus b a member of our span, of our
- column space, the span of these vectors?
- Well sure, what's a plus b? a plus b is equal to c1 plus b1
- times v1, plus c2 plus v2 times v2.
- I'm just literally adding this term to that
- term, to get that term.
- This term to this term to get this term.
- And then it goes all the way to Bn and plus Cn times Vn.
- Which is clearly just another linear
- combination of these guys.
- So this guy is definitely within the span.
- It doesn't have to be unique to a matrix.
- A matrix is just really just a way of writing a
- set of column vectors.
- So this applies to any span.
- So this is clearly a valid subspace.
- So the column space of a is clearly a valid subspace.
- Let's think about other ways we can interpret this notion
- of a column space.
- Let's think about it in terms of the expression-- let me get
- a good color-- if I were to multiply my-- let's think
- about this.
- Let's think about the set of all the values of if I take my
- m by n matrix a and I multiply it by any vector x, where x is
- a member of-- remember x has to be a member of Rn.
- It has to have n components in order for this multiplication
- to be well defined.
- So x has to be a member of Rn.
- Let's think about what this means.
- This says, look, I can take any member, any n component
- vector and multiply it by a, and I care about all of the
- possible products that this could equal, all the possible
- values of Ax, when I can pick and choose any
- possible x from Rn.
- Let's think about what that means.
- If I write a like that, and if I write x like this-- let me
- write it a little bit better, let me write x like this-- x1,
- x2, all the way to Xn.
- What is Ax?
- Well Ax could be rewritten as x1-- and we've seen this
- before-- Ax is equal to x1 times v1 plus x2 times v2, all
- the way to plus Xn times Vn.
- We've seen this multiple times.
- This comes out of our definition of
- matrix vector products.
- Now if Ax is equal to this, and I'm essentially saying
- that I can pick any vector x in Rn, I'm saying that I can
- pick all possible values of the entries here, all possible
- real values and all possible combinations of them.
- So what is this equal to?
- What is the set of all possible?
- So I could rewrite this statement here as the set of
- all possible x1 v1 plus x2 v2 all the way to Xn Vn, where
- x1, x2, all the way to Xn, are a member of the real numbers.
- That's all I'm saying here.
- This statement is the equivalent of this.
- When I say that the vector x can be any member of Rn, I'm
- saying that its components can be any
- members of the real numbers.
- So if I just take the set of all of the, essentially, the
- combinations of these column vectors where their real
- numbers, where their coefficients, are members of
- the real numbers.
- What am I doing?
- This is all the possible linear combinations of the
- column vectors of a.
- So this is equal to the span v1 v2, all the way to Vn,
- which is the exact same thing as the column space of A.
- So the column space of A, you could say what are all of the
- possible vectors, or the set of all vectors I can create by
- taking linear combinations of these guys, or the span of
- these guys.
- Or you can view it as, what are all of the possible values
- that Ax can take on if x is a member of Rn?
- So let's think about it this way.
- Let's say that I were to tell you that I need to solve the
- equation Ax is equal to-- well the convention is to write a b
- there-- but let me put a special b
- there, let me put b1.
- Let's say that I need to solve this equation
- Ax is equal to b1.
- And then I were to tell you-- let's say that I were to
- figure out the column space of A-- and I say b1 is not a
- member of the column space of A So what does that tell me?
- That tells me that this right here can never take on the
- value b1 because all of the values that this can take on
- is the column space of A.
- So if b1 is not in this, it means that this cannot take on
- the value of b1.
- So this would imply that this equation we're trying to set
- up, Ax is equal to b1, has no solution.
- If it had a solution, so let's say that Ax equals b2 has at
- least one solution.
- What does this mean?
- Well, that means that this, for a particular x or maybe
- for many different x's, you can definitely
- achieve this value.
- For there are some x's that when you multiply it by a, you
- definitely are able to get this value.
- So this implies that b2 is definitely a member of the
- column space of A.
- Some of this stuff on some level it's almost obvious.
- This comes out of the definition
- of the column space.
- The column space is all of the linear combinations of the
- column vectors, which another interpretation is all of the
- values that Ax can take on.
- So if I try to set Ax to some value that it can't take on,
- clearly I'm not going to have some solution.
- If I am able to find a solution, I am able to find
- some x value where Ax is equal to b2, then b2 definitely is
- one of the values that Ax can take on.
- Anyway, I think I'll leave you there.
- Now that you have at least a kind of abstract understanding
- of what a column space is.
- In the next couple of videos I'm going to try to bring
- everything together of what we know about column spaces, and
- null spaces, and whatever else to kind of understand a matrix
- and a matrix vector product from every possible direction.
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