Algebra

Rowspace and left nullspace | Matrix transformations | Linear Algebra | Khan Academy

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Suppose there is such a matrix A is a 2×3 matrix Treat this as a review Let’s calculate its zero space and column space Then the zero space of A is the set of all x They belong to – there are 3 columns here – So in R3 So that A is multiplied by this vector Equal to 0 vector We can build this collection like this I’m coming-we need to figure out All x satisfying this equation in R3 So take matrix A That is [2,-1,3;-4,2,6] Multiply by some arbitrary vector in R3 We get [x1;x2;x3] Make them equal to 0 vectors Is the 0 vector in R2 Because there are 2 rows Since a 2×3 matrix is \u200b\u200bmultiplied by a vector in R3 You get a 2×1 vector or 2×1 matrix So we get the 0 vector in R2 And to solve this system of equations – Is 2×1-x2-3×3=0 and many more We can write the augmented matrix We can build this augmented matrix as this Ie 2 -1 -3 -4 2 6 Then add a column we want to make it To solve this system of equations You know we are going to carry out a series of Row operation to make it into row simplified ladder shape And this will not change The right part of the augmented matrix This is the augmented matrix Therefore, the row of A simplifies the ladder-shaped null space Same as the zero space of A But anyway this is a review Let’s do some action Better to solve this Then first I want Divide the first row by 2 So if I divide the first row by 2 We get 1 -1/2 -3/2 Then of course dividing 0 by 2 is 0 Divide this line by – Divide by how many to simplify it- Divide by 4 So the first step is two row transformations You can do I could have done it in two steps So if you divide it by 4, it becomes -1 1/2 Then get 3/2 and 0 And now keep the first line unchanged I want to keep the first row unchanged Is 1 -1/2 -3/2 Of course 0 is on the right Now replace the second line with The second line plus the first line Is to perform linear action on them So -1+1=0 And 1/2+(-1/2)=0 And 3/2+(-3/2)=0 Of course 0+0=0 So what is left? We put it here This is another way to explain x1―― I write it like this-x1- I guess the easiest way to think about it is– Multiply the rows of A to simplify the ladder shape I.e. 1 -1/2 -3/2 Here is a string of 0 Multiplied by [x1;x2;x3] equals the 0 vector in R2 This is another explanation of augmented matrix That means this is useless This means 0 times this plus 0 times this Add 0 and multiply this is 0 So it tells us nothing But the first line tells us– I change a color– I.e. 1x1-1/2x2-3/2*x3 Is 0 A vector with all components satisfying this All in zero space If I want to write it differently I can write x1=1/2×2+3/2×3 Or if I want to write the solution set in vector form I can write zero space Are all vectors That is, the set of [x1;x2;x3] that meets these conditions what is this? Good x2 and x3 are free variables They are related to non-principal elements or It is related to the non-primary column in the row simplified ladder shape This is a main column I write like this Equal to x2 times something Plus x3 multiplied by something These are two free variables And we have x1 equal to 1/2×2 Is 1/2×2+3/2×3 And x2 is x2*1+0x3 And x3 is 0x2+1×3 Then zero space These can be any real numbers Free variables Then zero space is All linear combinations of this and this Or another way of writing it That is, the zero space of A is the space of Zhang Cheng This is the same as By the vector [1/2;1;0] and the linear combination of this vector Note that they are vectors in R3 This makes sense because zero space is Vector in R3 So this is Zhang Cheng’s space And this Zhangcheng space Then [3/2;0;1] like this So what is the column space of the original matrix A? The column space of A is equal to A subspace made up of linear combinations of all these things Or the space formed by the column vector Equal to [2;-4] And [-1;2] [-3;6] Zhang Cheng’s space These are different vectors So the space formed by these 3 vectors Now these may not be linearly independent In fact When turning it into a simplified ladder shape You know the basis vector for this is Vector related to the main column So here is a main column Is the first column So we can treat this as a basis vector It makes sense Because this is -2 times this thing This is -3/2 times this thing So these two things can be expressed as Linear combination of this So it is equal to the space of the vector [2;-4] So if you want to ask me And this is the basis of column space So if you want to know the rank– This is just a review Then the rank of A is equal to The number of vectors in the basis of our column space So equal to 1 Now all the calculations are just review But in the first few videos We have dealt with transposition So let’s do the calculations These quantities of the transposition of matrix A Then the transposition of A looks like this That is, the transpose of A is equal to 2 -1 -3 This is the first column Then the second column Yes -4 2 6 This is transpose Let’s do the math Transpose the zero space and column space of the matrix I turned it into a simplified ladder shape To get zero space Let me calculate the zero space of this We can make the same connection I write like this The zero space of the transposition of matrix A- The transpose of matrix A is a 3×2 matrix It is equal to all such vectors x They are in R2 Not in R3– Because now we take the transpose of the zero space – Multiply the transpose of A by the R vector Equal to the 0 vector in R3 We can calculate it in the same way as before We build an augmented matrix We can transform it into a simplified ladder shape Make them all equal to 0 Let’s do the math So if we– I turned it into a simplified ladder shape I divided the first row by 2 Divide the first row by 2 I want it to be simplified into a ladder shape The first line divided by 2 is 1 and -2 Then divide it by the second line – I want I want to keep it unchanged-it is -1 and 2 Then divide it by 3 in the last line Becomes -1 and 2 Now i want to keep the first row unchanged Namely 1 and -2 Now I want to change the second line to The second line plus the first line While -1+1=0 And 2+(-2)=0 Got some 0 I want to perform the same operation on the third row Replace it with it plus the first line Got some 0s again So this is the simplified ladder shape of the transposed lines of A And its zero space is the same as that of A’s transpose To find this zero space we can find all This equation. . . Multiply by vector That is, [x1;x2] is equal to the solution of [0;0;0] These are not vectors These are the numbers Is 0 0 and 0 So these two lines don’t tell us anything But this first line is useful We get 1×1-notice This is the main column here It is useful So x1 is a host variable And x2 is a free variable Clarify the first column Is our main column So if we go back to the transposition of A is the first column here Linked to the main column So when we talk about its column space This one needs to expand into a space This is also a review of what we have learned We want to apply it to transpose Back to zero space This tells us 1×1 I.e. x1-2×2=0 Maybe we can say x1=2×2 So all such vectors in R2 They meet these conditions In the zero space of A’s transpose I write like this That is, the zero space of A is The set of all such vectors – I write here-the set of all vectors Is [x1;x2] they are obvious in R2 Make x1 and x2 equal to – Well, our free variable is x2―― Is x2 times this vector So x1=2×2 Obviously x2――this is 2―― Equal to 1×2 So what is the result? This is all The linear combination of this vector here So we can say that it is equal to Vector [2;1] Zhang Cheng’s space Now this is zero space Sorry this is the zero space of A’s transposition I want to be very careful What is the column space now? The column space of matrix A’s transpose? The column space of a good A’s transposition is The set of all vectors in the space formed by the columns of A So you can say that the space formed by this column vector And this column vector But we know When we transform it into a simplified ladder shape Only this column of vectors Is related to the main column So this This is a linear combination of this If you multiply it by -2 You got this So it is consistent with what we already know So it is equal to this Zhang Cheng space That is the vector [2;-1;-3] This is a beautiful and elegant exercise Note that Zhang Cheng’s space is in R3 And it is the line in R3 Maybe in the next video I will make an icon of it But I have done the whole exercise Come to introduce you Transposed null space And transposed column space Think about the column space for transpose It is a subspace of the space formed by the vector-sorry- Is a subspace of the space formed by this vector and this vector This shows that this is a multiple of this So we can say that this vector But these are simplified steps of the original matrix A So we can think of it as The rows of the original matrix simplify the space formed by the stepped expansion This is this column it is The space formed by the column vectors of the transposed matrix Of course this is a linear combination of this So we can see it as The space formed by the columns of the transposed matrix It is equivalent to the subspace of the space formed by these lines Or we can call this the row space of A I write it down It is the column space of A’s transposition- This is the general situation I write out its general situation It’s not just for this example Then the column space of any matrix transpose This is called the row space of A This is a very natural name Because if A is a serial vector We can call it the transpose of some vectors Then this is the first line This is the second line Until the nth row like this These are the transpose of vectors They are row vectors If you imagine the space formed by these sheets From these row vectors it is Transposed column space Because when you transpose it Every vector becomes a column vector This is the meaning of line space Now transpose the zero space of the matrix Let’s write it like this– Are all vectors x satisfying this equation Equal to the 0 vector here Now if we What happens when you take the transpose of both sides of the equation? Well we already know the nature of transposition This is equal to Inverse product of transpose So this is equivalent to a vector Transpose of vector x If this is a column vector So now it becomes a row vector Then multiply the transpose of the transpose of A This is equal to Transpose of zero vector Or we can write like this We can write a certain matrix– Ok i write like this A certain column vector x―― What is the transpose of the transpose of matrix A? Is A So take the transpose of the column vector Now get the row vector You can think of it as a matrix If this is an element in Rn This is the 1×n matrix If this is the element in Rn Let’s change the order We multiply it by the transpose of the transpose We get matrix A We make it equal to the transpose of the zero vector Now this is very interesting We now have the equation for matrix A What is the null space of this matrix A? All vectors x in zero space Satisfy this equation equal to 0 So x is on the right So the zero space is all x that satisfies this The zero space of the transpose is All x satisfying this equation Then I say that all makes A transpose Multiply the set of x that equals 0 This is the zero space of A transpose Maybe we can write like this Written as all x makes x transpose multiplied by A equals Set of transposed zero vectors We call it another name This left zero space called A Why is it called the left zero space? Because now we are left multiplying x The general zero space is x on the right But now if you take the transposed zero space Use transposed properties This is equivalent to this transposed vector In fact I write Transpose here Multiply this transposed vector by A Multiply from the left So all x satisfying this Left zero space This is different from zero space Note the zero space of the transpose of A This is Zhang Cheng’s space This is also the left zero space of A What is the general null space of A? The general null space of matrix A is the plane in R3 This is the zero space of A The left null space of matrix A is the straight line in R2 This is different If you take space What is the row space of matrix A? The row space of matrix A is the straight line in R3 What is the column space of matrix A? Where did I calculate the null space of matrix A? Ok this is the only linearly independent vector It is a straight line in R2 So they are very different things Let’s take a look Their relationship Now there is something I want to tell you We figured it out The rank of this vector is 1 Because when you turn it into a simplified ladder shape Has a main column And the basis vector is Vector related to the main column Base vector This is the dimension of space So the dimension of the column space is 1. This is the same as rank What is the rank of A’s transposition now? The rank of the transpose of matrix A in this example When turning it into a simplified ladder shape Get a linearly independent column vector So the basis of the column space is equal to 1. Generally speaking, this will always appear The rank of matrix A Is the dimension of the column space Equal to the rank of the transpose of A If you think about this, it makes sense To calculate the rank of A you only need to calculate How many main columns Or another way How many principals When you are looking for the rank of the transposed vector You’re just saying-I know This is a little confusing– But when you want to calculate the rank of the transposed vector You mean how many of these column vectors Is it linearly independent? Or which ones are linearly independent? This is equivalent to asking How many rows are linearly independent? If you want to know how many column vectors are in the transposed matrix Is linearly independent This is equivalent to asking How many rows in the original matrix are linearly independent And when you turn this matrix into a row-simplified ladder shape Simplify everything in the ladder All work So they are Linear combination of the above vice versa All these things above are Linear combination So if you only have one pivot Then this thing itself Or a master of its own Can represent a basis of row space Or all rows can Represented by a linear combination of the main row Because of this you can count You can say good here is one The dimension of all row spaces is 1 This is the same as Dimension of the transposed column space I know this is a bit confusing It’s a little late for me now So lucky This makes you understand the rank of transposition The rank of the original matrix is \u200b\u200bthe same

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