NumPy Arrays and Linear Algebra

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Questions and Answers

What will the output of a.sum(axis=0) be for the array a = [[1, 2, 3, 4], [5, 6, 7, 8]]?

  • [3, 7, 11]
  • [36]
  • [10, 26]
  • [ 6, 8, 10, 12] (correct)

Which function would you use to find the maximum element in a NumPy array along a specific axis?

  • max() (correct)
  • argmax()
  • mean()
  • sum()

Which of the following methods would return the standard deviation of an entire NumPy array if no axis is specified?

  • sort()
  • mean()
  • std() (correct)
  • prod()

What does the lstsq() function in NumPy's linalg submodule return?

<p>The answer that minimizes the square error. (A)</p> Signup and view all the answers

What will the output of np.linalg.solve(A, b) be for the given matrices if A is invertible?

<p>The solution vector x of the equation Ax = b. (B)</p> Signup and view all the answers

Which function is used to compute the dot product of two matrices in NumPy?

<p>dot() (C)</p> Signup and view all the answers

What is the purpose of the inv() function in the linalg submodule?

<p>To return the inverse of a matrix. (B)</p> Signup and view all the answers

Which function would you use to compute the eigenvalues and eigenvectors of a given matrix in NumPy?

<p>eig() (D)</p> Signup and view all the answers

If A is a square matrix, what does the det() function compute?

<p>The determinant of the matrix. (C)</p> Signup and view all the answers

What will happen if argmax() is called on a NumPy array?

<p>It will return the index of the maximum element in the array. (D)</p> Signup and view all the answers

What happens when a value in a view array is modified?

<p>Both the view and original array are affected. (A)</p> Signup and view all the answers

How can you determine if an array is a view?

<p>Check if the base attribute is None. (C)</p> Signup and view all the answers

Which method can be used to create a copy of a view array?

<p>copy() (D)</p> Signup and view all the answers

Which of the following operations does not create a view?

<p>Using the copy method (C)</p> Signup and view all the answers

What will the line 'v = M[0,:]' return?

<p>The first row of the array M. (C)</p> Signup and view all the answers

What does the expression 'M.base' return if M is an original array?

<p>None. (B)</p> Signup and view all the answers

Which of the following correctly demonstrates changing a view without altering the original array?

<p>Use v = M.copy() and then modify v. (C)</p> Signup and view all the answers

When comparing arrays with '==' operator, what is a requirement for the arrays?

<p>They must be of the same shape or broadcastable. (A)</p> Signup and view all the answers

Which of the following best describes an array view?

<p>An array that shares data with another array. (B)</p> Signup and view all the answers

What are functions that act element-wise on NumPy arrays called?

<p>Universal functions (B)</p> Signup and view all the answers

Which function would give you the ceiling values of a NumPy array?

<p>np.ceil (C)</p> Signup and view all the answers

Using the vectorize decorator in NumPy allows you to convert which type of function?

<p>A function that acts on a single value (C)</p> Signup and view all the answers

What does the np.tan function do when applied to an array?

<p>Computes the tangent of each element in radians (A)</p> Signup and view all the answers

What would be the result of using np.exp on the array [1, 2, 3]?

<p>[2.71828183, 7.3890561, 20.08553692] (B)</p> Signup and view all the answers

Which of the following functions will convert an array to its floor values?

<p>np.floor (B)</p> Signup and view all the answers

The output of np.round([1.5, 2.5, 3.5]) will be?

<p>[2.0, 3.0, 4.0] (D)</p> Signup and view all the answers

Which of the following is an example of a universal function in NumPy?

<p>np.sqrt (C)</p> Signup and view all the answers

Flashcards

Universal Functions in NumPy

Universal functions in NumPy operate element-wise on arrays, applying the same function to each element. They are essential for performing mathematical operations efficiently on entire arrays.

Pre-Defined Universal Functions

NumPy provides a wide variety of pre-defined universal functions, including trigonometric functions (cos, sin, tan), exponential functions (exp), logarithms (log), rounding functions (ceil, floor, round), and many more.

Creating Your Own Universal Functions

You can create your own universal functions in NumPy using the vectorize function. This allows you to apply any Python function to each element of a NumPy array.

The vectorize Decorator

The vectorize decorator in NumPy provides a convenient way to automatically convert a regular Python function into a universal function.

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NumPy Array Views

A NumPy array view is a reference to the original array's data but with a different shape or perspective. It often facilitates efficient manipulation of data with minimal memory overhead.

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Boolean Arrays

Boolean arrays in NumPy are arrays where each element is a Boolean value (True or False). They are fundamental for filtering and selecting elements in arrays based on conditions.

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Array Indexing

Array indexing in NumPy allows you to access specific elements or subsets of an array using various methods, including integer indexing, slicing, and fancy indexing.

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Advanced Array Indexing

NumPy's advanced array indexing features, such as fancy indexing, allow you to select specific elements or subsets of an array using arrays of indices or Boolean conditions.

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sum(a, axis=None)

A NumPy array method for calculating the sum of all elements within a specified dimension, or across the entire array if no dimension is provided.

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min(a, axis=None)

A NumPy array method that returns the minimum value along a specific dimension or the minimum value across the entire array.

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prod(a, axis=None)

A NumPy array method that calculates the product of all elements within a specified dimension, or across the entire array if no dimension is provided.

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argmax(a, axis=None)

A NumPy array method that returns the index of the maximum value along a specific dimension, or across the entire array if no dimension is provided.

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argmin(a, axis=None)

A NumPy array method that returns the index of the minimum value along a specific dimension, or across the entire array if no dimension is provided.

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sort(a, axis=-1)

A NumPy array method that sorts the elements of the array along the axis dimension specified, or the last dimension if none is provided.

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argsort(a, axis=-1)

A NumPy array method that returns the indices that would sort the elements along the specified axis dimension.

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numpy.linalg

A NumPy submodule containing mathematical functions for performing linear algebra operations on matrices and vectors.

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numpy.linalg.solve(A, b)

A NumPy function within the linalg module that solves a system of linear equations defined by a non-singular matrix A and a corresponding vector b. Returns the solution vector x that satisfies Ax=b.

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numpy.linalg.svd(a)

A NumPy function within the linalg module that performs the singular value decomposition (SVD) on a matrix A. Returns a tuple containing the U, S, and V matrices of the decomposition.

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Invertible matrices

Invertible matrices have a unique inverse, allowing you to solve linear equations with a single solution.

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Least Squares Method ('lstsq')

The 'lstsq' function finds the best fit solution for equations that don't have exact solutions, minimizing the error.

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Array View

A view is a reference to an existing array's data, but with a different shape or perspective. Changes in the view affect the original data.

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Array 'base' Attribute

The 'base' attribute tells you if an array is original or a view. If 'base' is None, it's original.

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Slicing Arrays

Slicing an array creates a view, meaning changes in the slice affect the original array.

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Reshaping and Transpose

The 'reshape()' method and transpose operation create views, so changes in the new shape also affect the original data.

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Copying Array Views

Copying an array view creates an independent array, preventing changes from affecting the original.

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Array Comparisons

Comparison operations between arrays work element-wise, comparing corresponding entries. They also work on broadcastable arrays.

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Broadcastable Arrays

Arrays with different shapes can be compared if they are broadcastable - meaning their sizes are automatically adjusted.

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Study Notes

NumPy Arrays

  • NumPy arrays (ndarrays) offer various methods for operations on arrays, including mathematical functions.
  • Universal functions (ufuncs) operate element-wise on arrays; many are already defined in NumPy (e.g., cos, tan, exp, log, ceil, floor, round).

Creating Universal Functions

  • Python functions can be converted into universal functions using np.vectorize.
  • Alternatively, a decorator @np.vectorize can be applied directly above the function definition.

Array Methods

  • NumPy arrays (ndarray) have methods for specific operations (e.g., sum, min, max, mean, std, prod).
  • These methods often operate along a specified axis.
  • Function equivalents with the same name typically exist within the NumPy module. For instance, the sum function can also calculated by ndarray.sum()

Linear Algebra Functions

  • NumPy's linalg submodule provides linear algebra operations.
  • The solve function can solve systems of equations with invertible matrices ($Ax = b$).
  • For non-invertible matrices, lstsq (least squares) minimizes the square error.

Array Views

  • Array views share data with the original array.
  • Changes in a view affect the original array.
  • The base attribute of an array can distinguish between original arrays and views.
  • Operations like reshaping and transposing make views of the original array.

Boolean Operations

  • Boolean operations on arrays yield boolean arrays (e.g., comparisons using operators like ==, !=, <, >).
  • The all function checks if all elements are True; any checks if at least one element is True.

Integer Array Indexing

  • Integer arrays select elements from arrays;
  • Colons (:) in indices access all elements in the corresponding dimension.

Boolean Array Indexing

  • Boolean array indexing selects elements based on a boolean condition.
  • Boolean selection can be applied on multiple dimensions using boolean arrays with dimensions that correspond to the array shape.

Assignment by Array Indexing

  • The selected elements in arrays using indexing can be updated with new values (broadcastable).

The where Function

  • np.where(condition, x, y) can choose elements based on a condition.
  • If x and y are not given, the function returns indices where a condition is True.
  • If both x and y are given, the output returns values from x where a condition is True and y otherwise.

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