This x2 right here is x2 plusĢx3 plus 3x4 is equal to 0. What am I doing? I'm losing track of things. what's x1 equal to? It's equal to x3 timesġ plus x4 times 2. Of this, I could write x1, x2, x3, x4 is equal to Solution set to this equation, if I wanted to write it in terms I can solve this for x1 andĪ mistake here. Is equal to 0, and this obviously gives me no Here, there's no x1, you just have an x2, plus 2x3, plus 3x2 X3 minus x4, right? The 0 x2's is equal to 0. Here, this can be written as a system equations of x1 minus Has been reduced, just by doing reduce row echelonįorm, this problem. Row with the last row minus the middle row. It with the first row minus the second row, so IĬan get rid of this 1. Really just a little bit of an exercise just to Side of it, although these 0's are never going to change, it's Want to put this in reduced row echelon form, I want Guy with 4 times this guy, minus this guy. This 1 right here, so let me replace row 2 with rowĢ minus row 1.
Row echelon form, were actually just putting matrix A Multiply or subtract by, you're just doing it all timesĠ, so you just keep ending up with 0. Not going to change at all, because no matter what you When you put it into row echelon form, this right side's And to solve this, and we'veĭone this before, we're just going to put this augmented There, that's just the 0 vector right there.
We're going back to the augmented matrix world. This to equal that, and we wrote this as a system ofĮquations, but now we want to solve the system of equations, Should notice is we took the pain of multiplying this times I can represent this problemĪs the augmented matrix. We can represent this by anĪugmented matrix and then put that in reduced rowĮchelon form. Solution set to systems of equations like this. Solution set to this and we'll essentially have figured Vector with this column vector should be equal to that 0. Times this vector should be equal to that 0. Plus 2 times x2, plus 3 times x3, plus 4 times x4 is going And then this times this shouldīe equal to that 0. Plus 1 times x3, plus 1 times x4 is equal to this 0 there. The matrix multiplication, we get 1 times x1. So how do we do this? Well, this is just a straight Of these vectors is equal to the 0 vector. This particular A, such that my matrix A times any Vectors that are going to be members of R4, because I'm using Rn, but this is a 3 by 4 matrix, so these are all the Of all vectors that are a member of - we generally say Of all of these x's that satisfy this? Let me just write ourįormal notation. So I should have three 0's So myĠ vector is going to be the 0 vector in R3. Row times, that's the second entry, and then the third row. This and that's going to be the first entry, then this And what am I going to get? I'm going to have one row times The null space is the set ofĪll the vectors, and when I multiply it times A, I Only legitimately defined multiplication of this times aįour-component vector or a member of Rn. Times this vector I should get the 0 vector. X1, x2, x3, x4 is a member of our null space.
Times any of those vectors, so let me say that the vector Just the set of all the vectors that, when I multiply A But in this video let's actuallyĬalculate the null space for a matrix. Somewhat theoretically about what a null space isĪnd we showed that it is a valid subspace. There is much more to say but this should get you started thinking about it. If an nxn matrix A has n linearly independent row vectors the null space will be empty since the row space is all of R^n. When the row space gets larger the null space gets smaller since there are less orthogonal vectors. The dimension of a subspace generated by the row space will be equal to the number of row vectors that are linearly independent. Linear independence comes in when we start thinking about dimension. The orthogonal complement of the row space is the null space. In fact, given any subspace we can always find the orthogonal complement, which is the subspace containing all the orthogonal vectors. The vector x lives in the same dimension as the row vectors of A so we can ask if x is orthogonal to the row vectors. It is the subspace generated by the row vectors of A. The only way for Ax=0 is if every row of A is orthogonal to x.įrom this idea we define something called the row space. How do we compute Ax? When we multiply a matrix by a vector we take the dot product of the first row of A with x, then the dot product of the second row with x and so on. Recall that the definition of the nullspace of a matrix A is the set of vectors x such that Ax=0 i.e. The nullspace is very closely linked with orthogonality.
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