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We can create numpy arrays with more than one dimension.

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This section will focus only on 2D arrays but

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you can use numpy to build

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arrays of much higher dimensions.

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In this video, we will cover

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the basics and array creation in 2D,

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indexing and slicing in 2D,

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and basic operations in 2D.

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Consider the list a,

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the list contains three nested lists each of equal size.

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Each list is color-coded for simplicity.

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We can cast the list to a numpy array as follows.

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It is helpful to visualize the numpy array as

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a rectangular array each

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nested lists corresponds to

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a different row of the matrix.

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We can use the attribute ndim to obtain

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the number of axes or dimensions referred to as the rank.

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The term rank does not refer to the number of

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linearly independent columns like a matrix.

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It's useful to think of

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ndim as the number of nested lists.

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The first list represents the first dimension.

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This list contains another set of lists.

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This represents the second dimension or axis.

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The number of lists the list contains does not

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have to do with the dimension but the shape of the list.

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As with a 1D array,

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the attribute shape returns a tuple.

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It's helpful to use

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the rectangular representation as well.

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The first element in the tuple

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corresponds to the number of nested lists contained

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in the original list or the number of

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rows in the rectangular representation,

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in this case three.

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The second element corresponds to the size of each of

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the nested list or the number of

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columns in the rectangular array zero.

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The convention is to label this axis

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zero and this axis one as follows.

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We can also use the attribute size

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to get the size of the array.

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We see there are three rows and three columns.

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Multiplying the number of columns and rows together,

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we get the total number of elements,

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in this case nine.

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Check out the labs for arrays of

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different shapes and other attributes.

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We can use rectangular brackets to

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access the different elements of the array.

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The following image demonstrates the relationship between

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the indexing conventions for

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the lists like representation.

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The index in the first bracket corresponds to

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the different nested lists each a different color.

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The second bracket corresponds to the index

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of a particular element within the nested list.

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Using the rectangular representation,

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the first index corresponds to the row index.

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The second index corresponds to the column index.

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We could also use a single bracket

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to access the elements as follows.

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Consider the following syntax.

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This index corresponds to the second row,

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and this index the third column,

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the value is 23.

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Consider this example, this index corresponds to

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the first row and

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the second index corresponds to the first column,

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and a value of 11.

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We can also use slicing in numpy arrays.

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The first index corresponds to the first row.

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The second index accesses the first two columns.

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Consider this example,

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the first index corresponds to the first two rows.

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The second index accesses the last column.

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We can also add arrays,

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the process is identical to matrix addition.

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Consider the matrix X,

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each element is colored differently.

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Consider the matrix Y.

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Similarly, each element is colored differently.

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We can add the matrices.

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This corresponds to adding

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the elements in the same position,

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i.e adding elements contained

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in the same color boxes together.

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The result is a new matrix that has

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the same size as matrix Y or X.

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Each element in this new matrix is the sum of

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the corresponding elements in X and Y.

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To add two arrays in numpy,

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we define the array in this case X.

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Then we define the second array Y, we add the arrays.

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The result is identical to matrix addition.

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Multiplying a numpy array by

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a scalar is identical to

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multiplying a matrix by a scalar.

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Consider the matrix Y.

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If we multiply the matrix by this scalar two,

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we simply multiply every element in the matrix by two.

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The result is a new matrix of

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the same size where each element is multiplied by two.

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Consider the array Y.

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We first define the array,

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we multiply the array by a scalar as

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follows and assign it to the variable Z.

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The result is a new array

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where each element is multiplied by two.

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Multiplication of two arrays corresponds to

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an element-wise product, or Hadamard product.

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Consider array X and array

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Y. Hadamard product corresponds to multiplying each

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of the elements in the same position i.e

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multiplying elements contained in

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the same color boxes together.

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The result is a new matrix that is

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the same size as matrix Y or X.

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Each element in this new matrix is

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the product of the corresponding elements in X and Y.

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Consider the array X and Y.

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We can find the product of

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two arrays X and Y in one line,

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and assign it to the variable Z as follows.

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The result is identical to Hadamard product.

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We can also perform

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matrix multiplication with Numpy arrays.

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Matrix multiplication is a little more

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complex but let's provide a basic overview.

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Consider the matrix A

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where each row is a different color.

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Also, consider the matrix B

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where each column is a different color.

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In linear algebra, before we

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multiply matrix A by matrix B,

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we must make sure that the number of

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columns in matrix A in

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this case three is

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equal to the number of rows in matrix B,

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in this case three.

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From matrix multiplication, to obtain

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the ith row and jth column of the new matrix,

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we take the dot product of the ith row

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of a with the jth columns of B.

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For the first column,

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first row we take the dot product of

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the first row of A with the first column of B as follows.

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The result is zero.

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For the first row and the

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second column of the new matrix,

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we take the dot product of the first row of the matrix A,

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but this time we use the second column of matrix B,

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the result is two.

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For the second row and

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the first column of the new matrix,

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we take the dot product of

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the second row of the matrix A.

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With the first column of matrix B,

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the result is zero.

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Finally, for the second row

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and the second column of the new matrix,

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we take the dot product of the second row of

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the matrix A with the second column of matrix B,

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the result is two.

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In numpy, we can define the numpy arrays A and B.

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We can perform matrix multiplication and

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assign it to array C. The result is

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the array C. It corresponds to

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the matrix multiplication of array A and B.

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There is a lot more you can do with it in numpy.

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Checkout numpy.org.

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Thanks for watching this video.

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(Music)