6 Linear Algebra Solutions
title: "Linear algebra" date: 2026-05-24T13:51:52Z
Linear algebra
1import numpy as np
1np.__version__
'1.11.2'
Matrix and vector products
Q1. Predict the results of the following code.
1x = [1,2]
2y = [[4, 1], [2, 2]]
3print np.dot(x, y)
4print np.dot(y, x)
5print np.matmul(x, y)
6print np.inner(x, y)
7print np.inner(y, x)
[8 5]
[6 6]
[8 5]
[6 6]
[6 6]
Q2. Predict the results of the following code.
1x = [[1, 0], [0, 1]]
2y = [[4, 1], [2, 2], [1, 1]]
3print np.dot(y, x)
4print np.matmul(y, x)
[[4 1]
[2 2]
[1 1]]
[[4 1]
[2 2]
[1 1]]
Q3. Predict the results of the following code.
1x = np.array([[1, 4], [5, 6]])
2y = np.array([[4, 1], [2, 2]])
3print np.vdot(x, y)
4print np.vdot(y, x)
5print np.dot(x.flatten(), y.flatten())
6print np.inner(x.flatten(), y.flatten())
7print (x*y).sum()
30
30
30
30
30
Q4. Predict the results of the following code.
1x = np.array(['a', 'b'], dtype=object)
2y = np.array([1, 2])
3print np.inner(x, y)
4print np.inner(y, x)
5print np.outer(x, y)
6print np.outer(y, x)
abb
abb
[['a' 'aa']
['b' 'bb']]
[['a' 'b']
['aa' 'bb']]
Decompositions
Q5. Get the lower-trianglular L in the Cholesky decomposition of x and verify it.
1x = np.array([[4, 12, -16], [12, 37, -43], [-16, -43, 98]], dtype=np.int32)
2L = np.linalg.cholesky(x)
3print L
4assert np.array_equal(np.dot(L, L.T.conjugate()), x)
[[ 2. 0. 0.]
[ 6. 1. 0.]
[-8. 5. 3.]]
Q6. Compute the qr factorization of x and verify it.
1x = np.array([[12, -51, 4], [6, 167, -68], [-4, 24, -41]], dtype=np.float32)
2q, r = np.linalg.qr(x)
3print "q=\n", q, "\nr=\n", r
4assert np.allclose(np.dot(q, r), x)
q=
[[-0.85714287 0.39428571 0.33142856]
[-0.42857143 -0.90285712 -0.03428571]
[ 0.2857143 -0.17142858 0.94285715]]
r=
[[ -14. -21. 14.]
[ 0. -175. 70.]
[ 0. 0. -35.]]
Q7. Factor x by Singular Value Decomposition and verify it.
1x = np.array([[1, 0, 0, 0, 2], [0, 0, 3, 0, 0], [0, 0, 0, 0, 0], [0, 2, 0, 0, 0]], dtype=np.float32)
2U, s, V = np.linalg.svd(x, full_matrices=False)
3print "U=\n", U, "\ns=\n", s, "\nV=\n", v
4assert np.allclose(np.dot(U, np.dot(np.diag(s), V)), x)
U=
[[ 0. 1. 0. 0.]
[ 1. 0. 0. 0.]
[ 0. 0. 0. -1.]
[ 0. 0. 1. 0.]]
s=
[ 3. 2.23606801 2. 0. ]
V=
[[ 1. 0. 0.]
[ 0. 1. 0.]
[ 0. 0. 1.]]
Matrix eigenvalues
Q8. Compute the eigenvalues and right eigenvectors of x. (Name them eigenvals and eigenvecs, respectively)
1x = np.diag((1, 2, 3))
2eigenvals = np.linalg.eig(x)[0]
3eigenvals_ = np.linalg.eigvals(x)
4assert np.array_equal(eigenvals, eigenvals_)
5print "eigenvalues are\n", eigenvals
6eigenvecs = np.linalg.eig(x)[1]
7print "eigenvectors are\n", eigenvecs
eigenvalues are
[ 1. 2. 3.]
eigenvectors are
[[ 1. 0. 0.]
[ 0. 1. 0.]
[ 0. 0. 1.]]
Q9. Predict the results of the following code.
1print np.array_equal(np.dot(x, eigenvecs), eigenvals * eigenvecs)
True
Norms and other numbers
Q10. Calculate the Frobenius norm and the condition number of x.
1x = np.arange(1, 10).reshape((3, 3))
2print np.linalg.norm(x, 'fro')
3print np.linalg.cond(x, 'fro')
16.8819430161
4.56177073661e+17
Q11. Calculate the determinant of x.
1x = np.arange(1, 5).reshape((2, 2))
2out1 = np.linalg.det(x)
3out2 = x[0, 0] * x[1, 1] - x[0, 1] * x[1, 0]
4assert np.allclose(out1, out2)
5print out1
-2.0
Q12. Calculate the rank of x.
1x = np.eye(4)
2out1 = np.linalg.matrix_rank(x)
3out2 = np.linalg.svd(x)[1].size
4assert out1 == out2
5print out1
4
Q13. Compute the sign and natural logarithm of the determinant of x.
1x = np.arange(1, 5).reshape((2, 2))
2sign, logdet = np.linalg.slogdet(x)
3det = np.linalg.det(x)
4assert sign == np.sign(det)
5assert logdet == np.log(np.abs(det))
6print sign, logdet
-1.0 0.69314718056
Q14. Return the sum along the diagonal of x.
1x = np.eye(4)
2out1 = np.trace(x)
3out2 = x.diagonal().sum()
4assert out1 == out2
5print out1
4.0
Solving equations and inverting matrices
Q15. Compute the inverse of x.
1x = np.array([[1., 2.], [3., 4.]])
2out1 = np.linalg.inv(x)
3assert np.allclose(np.dot(x, out1), np.eye(2))
4print out1
[[-2. 1. ]
[ 1.5 -0.5]]