 In this video, we're going to solve question number one from the practice exam number three from math 2270 For which we're supposed we suppose that we have a three by five matrix that has a row space with Dimension equal to one. What is the nullity of the matrix? So some things we need to know in order to answer a question like this one So the end of the fundamental theorem of linear algebra sometimes referred to as the rank nullity theorem tells us the following If we take the rank of a matrix a and we add it to the nullity of the matrix a Which remind ourselves what these words here mean the rank of a matrix a this is the dimension of its column space It's also equal to the number of pivot columns in the matrix The nullity of a matrix is the dimension of its null space Which also equals the number of non pivot columns in the matrix or the number of free variables in the system We have if you have some in in excuse me in by in matrix here That is you have M mini rows and N mini columns then The rank of the matrix plus the nullity of the matrix is always add up to be N So the number the rank plus the nullity always adds up to the number of columns because after all the rank counts The number of pivot columns and the nullity counts the number of non pivot columns Related to this we also have the equation that the co-rank the co-rank of a Plus the co nullity of a This always adds up to M. Let's check what these vocab mean as well Co-rank this is the dimension of the row space and notice in our question. That's actually the information We have the row space is equal to one of the dimension the row space the co-rank is equal to one right here So the dimension of the row space is called the co-rank It's also equal to the number of pivot rows in the matrix The co-nullity this is equal to the dimension of the left no null space and hence is also equal to the number of non pivot rows inside of the matrix and so the sum of the Pivot rows plus the the sum of the non pivot rows will give you the number of rows in the matrix here's M So those are some important things to know here another important thing to know is that the rank in the co-rank are always Equal to each other and that's because after all the rank counts the number of pivot columns the co-rank counts the number of pivot rows And so That's just the same number Those are both the same number because this is the number of pivot positions inside of the matrix And so when we put these together what we need to find is The nullity of the matrix what we're given is the co-rank so we know the dimension of the row space We know this is equal to one so the way to connect these things together is the following we know that there are five Columns inside the matrix so we know n we also know that the rank is equal to one therefore The nullity by the rank knowledge theorem would have to equal five minus one which is equal to four And so therefore the correct answer would be e there the nullity of the matrix would necessarily have to be four