 Hello and welcome to the session. In this session we are going to discuss matrix representation of a vector, its components and multiplication of vector by a matrix. Now first of all we'll discuss vectors on coordinate plane. We know that a vector is a quantity which has both magnitude or length and direction. It is represented as directed segments such as vector AB. A vector on the coordinate plane can be described in component form or in terms of horizontal and vertical components where horizontal component is the x coordinate or the x component and the vertical component is the y coordinate or the y component. If we are given vector AB as 2 3 that is the ordered pair 2 3 on a coordinate plane then we'll represent vector AB as directed line which starts from the region that is the ordered pair 0 0 and ends at the ordered pair 2 3. We say that vector 2 3 has initial point as the ordered pair 0 0 and terminal point as the ordered pair 2 3. Now let us discuss matrix representation of vectors. A vector can be represented either in column matrix or in row matrix. So we can say that a matrix with dimension 1 cross 2 or 2 cross 1 can be used to represent a vector. So vector 2 3 can be written in column matrix form as a 2 by 1 matrix containing elements 2 3. Similarly vector 2 3 can be written in row matrix form as the 1 by 2 matrix containing elements 2 3. Mostly we write it in column matrix form. So vector 2 3 is written as a 2 by 1 matrix containing elements 2 3 with 2 is the x component and 3 represents y component. And now we'll see how to determine component form of a vector if vector A is given in the form of 2 by 1 matrix containing elements minus 3 2 and vector B is given as a 2 by 1 matrix containing elements 5 4. Then we need to express vector A B in component form and also we shall graph vector A B. Now x and y components of vector A B are found by finding B minus A. So B minus A is equal to B that is the 2 by 1 matrix containing elements 5 4 minus A that is a 2 by 1 matrix containing elements minus 3 2 and this is equal to a 2 by 1 matrix containing elements 5 minus of minus 3 4 minus 2 which is equal to a 2 by 1 matrix containing elements 5 minus of minus 3 is 5 plus 3 that is 8 4 minus 2 is 2. So B minus A is equal to a 2 by 1 matrix containing elements 8 2 or we can write it as vector A B which is given by B minus A is equal to a 2 by 1 matrix containing elements 8 2 where 8 is the x component and 2 is the y component. So its component vector form is the ordered pair 8 2. So we graph vector A B which is given by the ordered pair 8 2 it will start from the initial point that is the ordered pair 0 0 and will be extended to the point with the coordinates 8 2 which is the terminal point and we show the direction sign by this arrow. Now we shall discuss how to multiply a vector with a matrix. We have discussed vectors on coordinate plane having 2 components. Now vectors can also be 3 dimensional that is having 3 components and its matrix will have dimension 3 cross 1 or 1 cross 3. We know that we can multiply 2 matrices A and B if and only if the number of columns of matrix A is equal to number of rows of matrix B. Now suppose we have matrix A of order 2 cross 3 and we want to multiply it with a vector then the vector must have 3 rows in its column representation. It means vector will be 3 dimensional that is it will have 3 components. Now let us take an example. Suppose we have matrix A which is given by a 2 by 3 matrix containing elements in the first row as 1 2 3 and elements in the second row as 4 5 6 and if we want to find the product A B where B is a vector and number of rows in vector B is 3 and the order of this vector matrix is 3 cross 1. We know that we can multiply 2 matrices A and B if and only if the number of columns of matrix A is equal to number of rows of matrix B. Here we need to find the product A B and order of matrix A is 2 cross 3 and order of matrix B is 3 cross 1 and here we should note that number of columns of matrix A is equal to number of rows of matrix B. So we say that the product A B is defined and our resultant matrix will be of order 2 cross 1. So here we have a 2 by 1 matrix containing element in the first row as 2 obtain element in row 1 in the resultant matrix. We multiply elements of row 1 in matrix A with the corresponding elements of this column matrix that is matrix B and we have 1 into 1 plus 2 into 0 plus 3 into 1 to obtain element in the second row of the resultant matrix. We multiply elements in row 2 of matrix A with the corresponding elements of column 1 in matrix B and we will add the product and therefore we get 4 into 1 plus 5 into 0 plus 6 into 1 and this is equal to a 2 by 1 matrix containing element in the first row as 1 into 1 that is 1 plus 2 into 0 is 0 plus 3 into 1 is 3 and element in the second row as 4 into 1 is 4 plus 5 into 0 is 0 plus 6 into 1 is 6 and this is equal to a 2 by 1 matrix containing elements 1 plus 0 plus 3 that is 4, 4 plus 0 plus 6 that is 10. So product A B will be equal to a 2 by 1 matrix containing elements as 4, 10 so we get a new vector that is the ordered pair 4, 10 having 2 components. Thus in this session we have learnt matrix representation of a vector, its components and multiplication of vector by a matrix. This completes our session. Hope you enjoyed this session.