 So let's try to define some basic ideas in matrix arithmetic. Now if this wasn't an advanced math class, we could talk about matrix arithmetic in about 30 seconds. But linear algebra is an advanced math class, and so one of the goals of advanced math classes is learning to think like a mathematician. Mathematics typically proceeds in three stages. I've got an idea. Let's define equality and some operations. And then let's stretch these ideas to the breaking point and see what happens. So our current idea is that a linear transformation can be represented by a matrix T. So the question is given to transformation matrices A and B, can we define what we mean by A equals B, and can we define A plus B? So let's start with equality. Since a transformation matrix is defined by what it does to a vector, then A equals B means that A and B have to do the same thing to all vectors. Well if we consider the transformation matrix and what it actually does, this means that the only way we can have A equal to B is that all of the individual components of A must be the same as the individual components of B. In other words, we have to have component-wise equality. Now, you probably came to this conclusion on your own without thinking about it, but the important idea here is that we are basing our definitions on what these matrices do to vectors. And this will be important for addition and later multiplication. So let's consider addition. We might try to require that A plus B, whatever that is, acting on vector x, should be A acting on x plus B acting on x. And it's vitally important to remember there are only so many symbols. And in this particular case on the left-hand side, this plus refers to matrix addition, which we haven't yet defined. While on the right-hand side, the plus refers to vector addition, which we have. And this is useful because it suggests we can define matrix addition in terms of vector addition. And before proceeding, we'll note a few observations. So if A is a transformation matrix that takes vectors in fm and sends them to vectors in fn, and B is a transformation matrix, it has to be able to act on vector x. And so we know that B must take vectors in fm and send them to vectors in fp. But wait, there's more. Because we have to be able to add these two vectors afterwards, it's necessary that they have to have the same number of components. And so if B sends vectors in fm to fp, it's necessary that n and p be the same so that these vectors can be added. And that means that both A and B have to be n by m matrices. So suppose A is a transformation matrix that takes vectors in rm to vectors in rn. And we'll let our vector x have m components, x1 through xm. And remember, our transformation matrix gives us the coefficients of the linear formulas that give us the components of our vector y. So each row of ax gives the corresponding component of the transform vector. Our first component is going to be given by the formula. And similarly for the other components. Now this does lead to an important notational issue. The transformation matrix A, acting on the vector x, will produce a vector whose components would have to be written this way. And these are unreadably small. For readability, we can write it this way. We'll say that this vector, written as a column of entries, is a column vector. And you'll notice another important trait of thinking like a mathematician. We're not very creative when it comes up with creating names for objects. And we can go one step farther. If I apply the matrix A to the column vector x1 through xm, then I can write the result of that operation as A times our column vector gives us our column vector. And similarly, B acting on the same vector is going to give us another column vector. So our next step in trying to define addition is we'll let A plus B equal some matrix C, which will take vectors in RM and send them to vectors in RN. So we can write this as the matrix equation C applied to the column vector x1 through xm has to give us the column vector as shown. Now we want Cx to be equal to Ax plus Bx. So we need this output vector to be equal to our Ax vector plus our Bx vector. It's important to remember that this equality and this sum are based on vector equality and vector sum. So remember that vectors are added component wise and they are equal when their components are equal. Finally, remember the fact that we're writing our vectors as a column vector doesn't change the fact that they are vectors. So we'll compare our components. So our first component of our sum is going to be A1x1 plus A1 2x2 and so on plus our corresponding B values. And that should be the first component of our C vector. And we can rearrange this a little bit to get this equation. Now this has to be true for all vectors x. So the only way that we can have that equality for all values of xi is when the coefficients are equal. So A1 1 plus B1 1, the coefficient of x1, has to be equal to C1 1, the coefficient of x1. Likewise, A1 2 plus B1 2 must be equal to C1 2 and so on. And so this suggests the following definition of addition of two matrices. Let A and B be two matrices of the same size. Then A plus B is going to be the matrix C, where the components of C are the sum of the corresponding components of A and B. And like vector addition and vector equality, we're going to define matrix addition using component wise addition. And so let's put down as our exciting example the sum of the two matrices. And we'll find our sum by adding our entries component wise. So to get the entry in the first row, first column, we'll add the entries in the first row, first column of the two matrices. So that's going to be 2 plus 1 gives us 3. Likewise, first row, second column, we'll add the first row, second column entries, 5 plus 1 equals 6, and first row, third column, we'll add the first row, third column entries, 1 plus 4 equals 5. And we'll do that for all the other entries and get our sum of the two matrices.