Hi,
Now that we have seen a little about matrices, let's now discuss about matrix multiplication. I will not be discussing how matrix multiplication is done. That is essentially a method. However, i will try to express what matrix multiplication essentially is. Please correct me if there are mistakes in my posts.
Now, a matrix is essentially a bunch of vectors. We can visualize a matrix in two ways.
a) Matrix as a bunch of row vectors:$${\begin{pmatrix} \vec r_1 \\ \vec r_2 \end{pmatrix}}$$
Here,$$ \vec r_1 = (a_1, a_2, a_3..a_n) $$$$ \vec r_2 = (b_1, b_2, b_3..b_n) $$
Thus, the matrix is an 2 x n matrix. since it has two rows and n columns.
b)Matrix as a bunch of column vectors
Now, the same matrix can be expressed in another way:$${\begin{pmatrix} \vec c_1 \vec c_2 .... \vec c_n \end{pmatrix}}$$
where $$c_1 = (a_1, b_1)$$$$c_2 = (a_2, b_2)$$$$c_n = (a_n, b_n)$$
Thus, we have two ways of looking at the same matrix!Now, let's take a look at matrix multiplication. A matrix multiplication with a vector is essentially the dot product of the vector with each of the row vectors of the matrix. Or, it can also be seen as the sum of the scalar products or the column vectors.
Let's take an example. Let's take a 2x2 matrix $$A = {\begin{pmatrix} 1 2 \\ 3 4 \end{pmatrix}}$$ or
$$A = {\begin{pmatrix} \vec r_1 \\ \vec r_2 \end{pmatrix}}$$
where $$ \vec r_1 = (1, 2)$$$$\vec r_2 = (3, 4)$$
Now, let $$\vec x = (x_1, x_2) $$Now, A.X can be written as $${\begin{pmatrix} \vec r_1 dot \vec x\\ \vec r_2 dot \vec x \end{pmatrix}}$$The same can also be written as:$$x_1 * \vec c_1 + x_2 * \vec c_2$$
where c1 and c2 are column vectors of A, i.e c1 = (1, 3) and c2 = (2, 4).Thus, this is an interesting way in which matrices can be looked at.
Now that we have seen a little about matrices, let's now discuss about matrix multiplication. I will not be discussing how matrix multiplication is done. That is essentially a method. However, i will try to express what matrix multiplication essentially is. Please correct me if there are mistakes in my posts.
Matrix Multiplication by a vector
Before we dive into multiplication of matrices with vectors, let's closely examine what truly a matrix is.Now, a matrix is essentially a bunch of vectors. We can visualize a matrix in two ways.
a) Matrix as a bunch of row vectors:$${\begin{pmatrix} \vec r_1 \\ \vec r_2 \end{pmatrix}}$$
Here,$$ \vec r_1 = (a_1, a_2, a_3..a_n) $$$$ \vec r_2 = (b_1, b_2, b_3..b_n) $$
Thus, the matrix is an 2 x n matrix. since it has two rows and n columns.
b)Matrix as a bunch of column vectors
Now, the same matrix can be expressed in another way:$${\begin{pmatrix} \vec c_1 \vec c_2 .... \vec c_n \end{pmatrix}}$$
where $$c_1 = (a_1, b_1)$$$$c_2 = (a_2, b_2)$$$$c_n = (a_n, b_n)$$
Thus, we have two ways of looking at the same matrix!Now, let's take a look at matrix multiplication. A matrix multiplication with a vector is essentially the dot product of the vector with each of the row vectors of the matrix. Or, it can also be seen as the sum of the scalar products or the column vectors.
Let's take an example. Let's take a 2x2 matrix $$A = {\begin{pmatrix} 1 2 \\ 3 4 \end{pmatrix}}$$ or
$$A = {\begin{pmatrix} \vec r_1 \\ \vec r_2 \end{pmatrix}}$$
where $$ \vec r_1 = (1, 2)$$$$\vec r_2 = (3, 4)$$
Now, let $$\vec x = (x_1, x_2) $$Now, A.X can be written as $${\begin{pmatrix} \vec r_1 dot \vec x\\ \vec r_2 dot \vec x \end{pmatrix}}$$The same can also be written as:$$x_1 * \vec c_1 + x_2 * \vec c_2$$
where c1 and c2 are column vectors of A, i.e c1 = (1, 3) and c2 = (2, 4).Thus, this is an interesting way in which matrices can be looked at.
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