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In class We made the statement that the matrix ST associatedl wich full rankreparaeterization has thc SAme Fl space AS the matrix X Prove this state-MICIt is (rule,...

Question

In class We made the statement that the matrix ST associatedl wich full rankreparaeterization has thc SAme Fl space AS the matrix X Prove this state-MICIt is (rule,

In class We made the statement that the matrix ST associatedl wich full rank reparaeterization has thc SAme Fl space AS the matrix X Prove this state- MICIt is (rule,



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Prove that if $A$ is a matrix whose nullspace and column space are the same, then $A$ must have an even number of columns.

All right. All right, So we have some arbitrary upper triangular matrix A. With some entries A one down to an in the diagonal, some arbitrary entries in the upper trip and the upper part and all zero below the diagonal for B I J. We're going to use uh we're going to assume for first we're just going to assume that jay is actually less than I. And then we take out the column J right here, which means that all these entries, all the entries to the right of that get moved over. And so this and so then this, once this line right here gets shifted over and then there and then we do the same thing for the I throw for the I throw here, which means that every entry down here gets moved up. And so we get something that looks like this where there's actually these entries right here that are below the diagonal. And so this is not an upper triangular matrix. Okay, now we're going to assume that B i J when i is less than jay? Oh so we get so we take out so we take out some we're going to take out some upper row right here. Yeah. And then that means that a one dot dot dot all entries over here get deleted. So everything below here gets moved up, we get then get zero here and then this move and then we get these arbitrary entries up here actually it's going to look more like this. So and then we take out some J column right here, which means that all the entries to the right of this column get moved over and then we have zero here and move and then we and then we keep going this way. two a.m. Yeah. Mhm. So you can see that and then there's just gonna be the same thing, There's gonna be some arbitrary entries right over here. Non non zero arbitrary entries. And so we and so since everything here below the diagonal is still zero, this means b i J is an upper triangular matrix.

Hello there. So let's suppose that we have a matrix that is square matrix. And we need to prove that the determinant of a transfer a transposed A. Is equal to the determinant of a transposed. We can directly proved this by the properties of the attorney in that say that the multiplication of A. B is equal to be A. That will be enough. But let's prove this in a different way. It really works as well. So we have that the turbulent of 80 a. By the properties of the terminal will be equal to the multiplication of the terminal of a transposed determinant A. And this is just because these are numbers the determinant will return and real real value. We can interchange them. So we obtained that this is equal to determinant of a determinant of 80 and this is just a determinant of a. T. Us required another way to prove this. That is also very simple is that let's say that B is equal to a T. A. Then we saw that the transpose of tea that is a transposed A transposed is equal two. The tories. Not in this way, sorry, I have a mistake here. Yeah. And the other way is saying that we have the A. Is the cost to a transposed and B is the cost to aim. I am. By the properties of determinant, we know that determinant of a. B is equal to determinant of B. A. Uh when we replace the values of A. And B, we have the tournament of 88 Equals two determinant of a 80. As required. You can prove it In either of the Tuesday's two ways.


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