<jupyter_start><jupyter_text>Practice1. Given the linear combination below, try to create a corresponding matrix representing it.$$\theta = 5x + 3y - z$$<jupyter_code>import numpy as np

Θ = np.array([5, 3, -1])
print(Θ)<jupyter_output>[ 5  3 -1]<jupyter_text>2. Given the system of linear combinations below, try to encode it as a matrix. Also describe the matrix.$$ A = \left\{\begin{array} 5x_1 + 2x_2 +x_3\\ 4x_2 - x_3\\ 10x_3 \end{array}\right. $$<jupyter_code>def describe_mat(matrix):
    print(f'Matrix:\n{matrix}\n\nShape:\t{matrix.shape}\nRank:\t{matrix.ndim}\n')
A = np.array([
    [1, 2, 1],
    [0, 4, -1],
    [0, 0, 10]
])
describe_mat(A)<jupyter_output>Matrix:
[[ 1  2  1]
 [ 0  4 -1]
 [ 0  0 10]]

Shape:	(3, 3)
Rank:	2<jupyter_text>3. Given the matrix below, express it as a linear combination in a markdown.<jupyter_code>G = np.array([
    [1,7,8],
    [2,2,2],
    [4,6,7]
])
G = np.diag(G)
G<jupyter_output><empty_output><jupyter_text>$$ x + 2y + 7z$$  4. Given the matrix below, display the output as a LaTeX makdown also express it as a system of linear combinations.<jupyter_code>H = np.tril(G)
H<jupyter_output><empty_output>
{'fraction_non_alphanumeric': 0.1608695652173913, 'fraction_numerical': 0.049565217391304345, 'mean_word_length': 4.092920353982301, 'pattern_counts': {'":': 0, '<': 15, '<?xml version=': 0, '>': 15, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
{'chain_length': '4', 'dir': 'jupyter-structured-clean-dedup', 'id': '242203', 'license': 'Apache-2.0', 'n_tokens_mistral': 501, 'n_tokens_neox': 471, 'n_words': 130, 'path': 'Ricafrente_met.ipynb', 'repo_name': 'Dayle227/Linear-Algebra', 'size': '1150'}