Q2

In this question, you'll implement main methods that are crucial for linear algebra: vector and matrix multiplication.

Part A

Write a function which:

is named mv_multiply
takes 2 arguments: a 2D NumPy matrix as the first argument, and a 1D NumPy vector as the second
returns 1 value: a NumPy array containing the product of the multiplication

This function will specifically perform the operation $\vec{y} = A * \vec{x}$, where $A$ and $\vec{x}$ are the function arguments. Remember how to perform this multiplication:

First, you need to check that the number of columns of $A$ is the same as the length of $\vec{x}$. If not, you should print an error message and return None.
Second, you'll compute the dot product of each row of $A$ with the entire vector $\vec{x}$. This can be done using the numpy.dot function.
Third, the result of the dot product from the $i^{th}$ row of $A$ will go in the $i^{th}$ element of the solution vector, $\vec{y}$. Therefore, $\vec{y}$ will have the same number of elements as rows of $A$.

You can use numpy.array and numpy.dot functions, and range(), but no other functions.



In [ ]:



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import numpy as np
np.random.seed(487543)

A = np.random.random((92, 458))
v = np.random.random(458)
np.testing.assert_allclose(mv_multiply(A, v), np.dot(A, v))



In [ ]:

    
import numpy as np
np.random.seed(49589)

A = np.random.random((83, 75))
v = np.random.random(83)
assert mv_multiply(A, v) is None

Part B

Write a function which:

is named mm_multiply
takes two 2D NumPy matrices as arguments
returns 1 2D NumPy array: the product of the two input matrices

This function will perform the operation $Z = X \times Y$, where $X$ and $Y$ are the function arguments. Remember how to perform matrix-matrix multiplication:

First, you need to make sure the matrix dimensions line up. For computing $X \times Y$, this means the number of columns of $X$ (first matrix) should match the number of rows of $Y$ (second matrix). These are referred to as the "inner dimensions"--matrix dimensions are usually cited as "rows by columns", so the second dimension of the first operand $X$ is on the "inside" of the operation; same with the first dimension of the second operand, $Y$. If the operation were instead $Y \times X$, you would need to make sure that the number of columns of $Y$ matches the number of rows of $X$. If these numbers don't match, you should return None.
Second, you'll need to create your output matrix, $Z$. The dimensions of this matrix will be the "outer dimensions" of the two operands: if we're computing $X \times Y$, then $Z$'s dimensions will have the same number of rows as $X$ (the first matrix), and the same number of columns as $Y$ (the second matrix).
Third, you'll need to compute pairwise dot products. If the operation is $X \times Y$, then these dot products will be between the $i^{th}$ row of $X$ with the $j^{th}$ column of $Y$. This resulting dot product will then go in Z[i][j]. So first, you'll find the dot product of row 0 of $X$ with column 0 of $Y$, and put that in Z[0][0]. Then you'll find the dot product of row 0 of $X$ with column 1 of $Y$, and put that in Z[0][1]. And so on, until all rows and columns of $X$ and $Y$ have been dot-product-ed with each other.

You can use numpy.array, but no functions associated with computing matrix products (and definitely not the @ operator).

You CANNOT use numpy.dot, because that can be used to explicitly compute matrix-matrix products. You can only use it indirectly by using your mv_multiply function from Part A.



In [ ]:



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import numpy as np
np.random.seed(489547)

A = np.random.random((48, 683))
B = np.random.random((683, 58))
np.testing.assert_allclose(mm_multiply(A, B), A @ B)



In [ ]:

    
np.random.seed(466525)

A = np.random.random((58, 683))
B = np.random.random((683, 58))
np.testing.assert_allclose(mm_multiply(B, A), B @ A)



In [ ]:

    
A = np.random.random((359, 45))
B = np.random.random((83, 495))
assert mm_multiply(A, B) is None