Q2

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

A

You've done this before, but you'll do it once more: write a function named dot_product which takes two 1D NumPy arrays (vectors) and computes their pairwise dot product. The function will take two arguments: two 1D NumPy arrays. It will return one floating-point number: the dot product.

Recall that the dot product is a sum of products: the corresponding elements of two vectors are multiplied with each other, then all those products are summed up into one final number.

For example, dot_product([1, 2, 3], [4, 5, 6]) will perform the operation:

(1 * 4) + (2 * 5) + (3 * 6) = 4 + 10 + 18 = 32

so dot_product([1, 2, 3], [4, 5, 6]) should return 32.

If the vectors are two different lengths, this function should return None.

You can use numpy for arrays and the numpy.sum function, but no others (especially not the numpy.dot function which does exactly this).

B

Write a function mv_multiply which takes a 2D NumPy matrix as the first argument, and a 1D NumPy vector as the second, and multiplies them together, returning the resulting vector.

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 $\vec{x}$.

• 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 for arrays, and your dot_product function from Part A, but no other functions.

C

Write a function mm_multiply which takes two 2D NumPy matrices as arguments, and returns their matrix product.

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, but no functions associated with computing matrix products (and definitely not the @ operator).

Hint: you can make use of your mv_multiply and/or dot_product methods from previous questions to help simplify your code.