[BOŞ_SES] Now The order of the matrix and the solution that we have seen the impact, Coming now to the practical work so much easier to coefficients We are looking at the order of the matrix extended We are looking at these two are equal to the order of the matrix solution The only solution to this same order which equals the number of unknowns, unknown number not equal There are infinitely many solutions at that time. Now here we see three instances, these three examples Gaussian We will find the qualifying order, scroll. In this example, we have solved well before 80 to 87. In between pages 92 and 97 and even as a Gauss Jordan. We solved the page. The first equation as follows: that's a x x two from If the coefficient matrix coefficients only purify the summer here, attaching the right side next to it is understood that the extended matrix. When applying the Gaussian elimination, all the details on this 80 to 87. There on the page, we reach the following equation, we arrive at the following structure; here three in the first column because this column as it appears the number of independent two zero because it is impossible to produce in any way a. It is impossible to produce using two last in the last column, so here are three independent column. It is readily apparent in three independent lines but this It does not matter for our judgment. Now these three columns to the right in attaching now been dependent on the four pillars of independent Is? This independent, sorry, because a three-dependent component When we can express each of these three vectors of a vector, so we are in three-dimensional space of the three-dimensional space a base assembly, these vectors, because the columns are independent vectors. On the right side the third, three component, can always be expressed as a vector in three dimensions because they, hence the fourth row in the right-hand side, To add a column does four independent vectors. Thus, the matrix coefficients extended column of the matrix Number three, which equals the number of unknowns. That three of these columns to be equal, regardless of the number shows that the solution be equal, it also shows that the number is equal to the unknown because it is the only solution In order for us to independently therefore, the order of columns gives access to space. We know the definition of the famous theorem The size of the space, then the number of unknowns, The size of the equal access to space, that it is the number of columns, The size of the space plus zero. The number of this column with the unknown The order of zero is zero and that space would be one solution for the same. Indeed, this solution is the only solution of the equation is that this team earlier We found in the pages, where X is a one hundred twenty four x two of seventy-five, x If it is seen as a place where thirty-three of them provided all three equations. In the second equation, we have the following structure, yes again in the third equation x one x two are x remove three of coefficients constitute the matrix, The right side of the matrix by adding extended If we apply this form and Gaussian elimination, It was again a problem we solved this before, It is seen in the detail page 92 and 97, wherein like structure reached, refer to the order of the matrix two because the third row is always zero this right to the exit of the third column first shown by two, The third column that depends on the first two. But the right As you can see we consider the matrix of zeros place a number of these thirty thirty-one came with bir'l You can not produce what components are received If you receive this column. Thus far right column of the matrix It is independent of the column that is outside the space shuttle. Therefore there is no solution. Indeed, if we look at this solution is not a solution that we see, zero plus zero times x times x times x three two plus zero zero all data, that is equal to zero leads to a contradiction, thirty-one. This enables the equation no x a, x two, x indicates three that indicates whether that solution. Here we see the following; the coefficient matrix We see two, but three of the order is the order of the extended matrix, This solution also said that according to us immediately theorem. We already see this in the traditional structure, like a thirty equals zero we're getting inconsistent results. Our third equation seemed that here again it said We solve this equation with Gaussian elimination pages. x of the coefficient matrix and the right side purify You would expect to obtain by adding the expanded matrix structure Gauss application, we apply the Gaussian elimination, The last line is always zero, as seen here building reached. Now when we look at the matrix coefficients but there are two free-standing columns, The last element of this right is attaching the right side, In this example the first two to the last component of the others zero is a vector dependent from column to column. Judging by the more abstract the number of independent coefficients of two columns of the matrix, hence the extent of his two matrix coefficients of two extended. There are solutions for this is equal to two orders of magnitude but unknown number three, therefore null space, there is a zero-dimensional space, so here is a solution, any solution at zero space When you add a solution still happening. In fact, we do it with the Gaussian elimination method x üç'et we say, x t, we see two plus two is three. They took away If we get a place in the first equation for x. These values, as seen in an unknown random t remains to be selected, whatever we choose t again turns out to be a solution, infinite solutions and therefore we see that this only we can determine by looking at the order. We finish this chapter, here are several homework questions. These are questions that we did before. We want to identify them, as given here right sides whether the solution is received, If you have one, or is that an infinite number We want us to find. And here, in these equations, two variable equations, we can also solve them all by hand. Including three unknowns equation Looking at just order We want to understand the nature of the solution. This section is finished again, as we did in each section work begins with the representation of the matrix When we look at the summary of the vector x, Displaying the definition of space vector x to base vector, How's that for a matrix with target space include the right column (e1) A (e2), A (ej) s and placing them in terms of foundations, so we can find the terms of the vector in the target space. And yet it summarizes as usual looking at what can occur easily if you know what. Bye now, this section whereby transformation matrices were able to show, I can say that some operations in the matrix,
