Linear Algebra: Vectors and Matrices
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Questions and Answers

What is the purpose of the show_vector function in the given Python code?

  • To calculate the magnitude of a vector
  • To compute the dot product of two vectors
  • To visualize a vector in a 2D plot (correct)
  • To perform linear interpolation between two vectors

What is the result of the operation pos = pos + vel in the given GLSL code?

  • pos is scaled by the value of vel
  • pos is translated by the value of vel (correct)
  • pos is rotated by the value of vel
  • pos is set to the difference between pos and vel

What is the purpose of the np.linspace function in the given Python code?

  • To compute the dot product of two vectors
  • To generate a range of random numbers
  • To generate a range of evenly spaced numbers over a specified interval (correct)
  • To create a matrix with specified dimensions

What does the np.linalg.norm function compute?

<p>The magnitude of a vector (A)</p> Signup and view all the answers

What is the purpose of the ax.set_aspect(1.0) line in the given Python code?

<p>To ensure the aspect ratio of the plot is preserved (B)</p> Signup and view all the answers

What is the effect of the line show_vector(ax, origin, alpha * v1 + (1 - alpha) * v2, color='k', alpha=0.25, label='')?

<p>It displays a weighted sum of v1 and v2 (B)</p> Signup and view all the answers

What is the purpose of the ax.set_ylabel() function in the given Python code?

<p>To set the y-axis label (D)</p> Signup and view all the answers

What is the function of the scipy.special.gamma() function in the given Python code?

<p>To calculate the gamma function (D)</p> Signup and view all the answers

What is the purpose of the ax.set_title() function in the given Python code?

<p>To set the title of the plot (D)</p> Signup and view all the answers

What is the purpose of the plt.hist2d() function in the given Python code?

<p>To create a 2D histogram (A)</p> Signup and view all the answers

What is the purpose of the np.arange() function in the given Python code?

<p>To create an array of numbers (B)</p> Signup and view all the answers

What is the value of 'd' in the given experiment?

<p>1 (D)</p> Signup and view all the answers

What is the purpose of the line 'points_in_sphere = points_in_box[np.linalg.norm(points_in_box,axis=1) < 1]'?

<p>To select points with radius &lt; 1 (inside a sphere in that cube) (B)</p> Signup and view all the answers

What is the condition for matrix multiplication in the provided 'matmul' function?

<p>The number of columns in the first matrix must match the number of rows in the second matrix (A)</p> Signup and view all the answers

What is the result of the matrix multiplication 'matmul(a, b)' in the provided example?

<p>[[-4. 4.]] (D)</p> Signup and view all the answers

What is the purpose of the assert statement in the 'matmul' function?

<p>To check if the matrices can be multiplied (D)</p> Signup and view all the answers

What does the line 'c[i, j] = np.sum(a[i, :] * b[:, j])' in the 'matmul' function do?

<p>It calculates the weighted sum of the ith row of A and the jth column of B (D)</p> Signup and view all the answers

What is the order of operations in matrix A?

<p>Rotate, then scale (A)</p> Signup and view all the answers

What is the difference between matrices A and B?

<p>They represent different orders of scaling and rotation (B)</p> Signup and view all the answers

What is the purpose of the line mu = np.mean(x, axis=0)?

<p>To compute the mean of the dataset (C)</p> Signup and view all the answers

What does the @ symbol represent in the line (x - mu).T @ (x - mu)?

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

What is the purpose of the np.set_printoptions function?

<p>To set the precision of printed numbers (D)</p> Signup and view all the answers

What is the relationship between sigma_cross and sigma_np?

<p>They are identical (D)</p> Signup and view all the answers

What is the function of the eigsorted function in the given Python code?

<p>To sort the eigenvalues and eigenvectors of a covariance matrix (A)</p> Signup and view all the answers

What is the purpose of the np.cov function in the given Python code?

<p>To calculate the covariance of a given dataset (C)</p> Signup and view all the answers

What is the result of the operation np.eye(3) @ np.array([4,5,6])?

<p>[4, 5, 6] (A)</p> Signup and view all the answers

What is the purpose of the np.fliplr function in the given Python code?

<p>To calculate the anti-diagonal of an array (C)</p> Signup and view all the answers

What is the result of the operation aI = np.eye(3) * a?

<p>A matrix with diagonal elements equal to a and other elements equal to 0 (C)</p> Signup and view all the answers

What is the purpose of the ax.set_aspect(1.0) line in the given Python code?

<p>To set the aspect ratio of the plot to be equal (A)</p> Signup and view all the answers

Study Notes

Vectors and Vector Operations

  • Vectors can be added and scaled using simple operations
  • The show_vector function is used to visualize vectors in a 2D plane
  • Vectors can be interpolated using linear interpolation, which is a weighted sum of two vectors
  • The np.linalg.norm function is used to calculate the magnitude of a vector

Matrix Operations

  • Matrices can be multiplied using the @ operator or the matmul function
  • Matrix multiplication is only possible if the number of columns in the first matrix matches the number of rows in the second matrix
  • The matmul function can be used to perform matrix multiplication with broadcasting
  • The np.transpose function can be used to transpose a matrix

Linear Transformations

  • Linear transformations can be represented as matrices
  • The matrix product of two linear transformations is a new linear transformation
  • The order of matrix multiplication matters (e.g. scale_x @ rot30 is not the same as rot30 @ scale_x)
  • Linear transformations can be visualized using matrices and scatter plots

Covariance and Correlation

  • The covariance of a dataset can be calculated using the np.cov function
  • The covariance matrix represents the variance and covariance of each pair of variables
  • The eigsorted function is used to sort the eigenvalues and eigenvectors of a covariance matrix
  • The cov_ellipse function is used to visualize the covariance ellipse of a dataset

Diagonal and Identity Matrices

  • Diagonal matrices can be created using the np.diag function
  • The identity matrix can be created using the np.eye function
  • The identity matrix does not change the result when multiplied with another matrix
  • Diagonal and identity matrices have specific properties when multiplied with other matrices

Miscellaneous

  • The np.random.normal function can be used to generate random normal variables
  • The plt.hist2d function can be used to create a 2D histogram
  • The scipy.special.gamma function is used to calculate the gamma function

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Description

Learn about vector operations, including addition, scaling, and interpolation, as well as matrix operations, including multiplication. Functions like show_vector and np.linalg.norm are also covered.

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