couple of notebook added for review

linear_algebra/02_linear_transformations.py

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+import marimo
+
+__generated_with = "0.15.5"
+app = marimo.App()
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        # 2. Linear Transformations: Intuition & Examples
+
+        Linear transformations are operations that move, rotate, scale, or shear vectors and shapes in space. They are fundamental in machine learning for manipulating data and features.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Example 1: Rotating a Vector 0001F504
+        - **Original vector:** $v = [2, 1]$
+        - **Rotation:** 45° counterclockwise
+        - **Transformation matrix:**
+          $$ R = egin{bmatrix} os	heta & -in	heta \ in	heta & os	heta nd{bmatrix} $$
+        - **Result:** Vector is rotated in space
+        """
+    )
+    return
+
+
+@app.cell
+def _():
+    import numpy as np
+    import matplotlib.pyplot as plt
+
+    v = np.array([2, 1])
+    theta = np.pi / 4
+    R = np.array([[np.cos(theta), -np.sin(theta)], [np.sin(theta), np.cos(theta)]])
+    v_rot = R @ v
+
+    plt.figure(figsize=(6,6))
+    plt.quiver(0, 0, v[0], v[1], angles='xy', scale_units='xy', scale=1, color='r', label='Original')
+    plt.quiver(0, 0, v_rot[0], v_rot[1], angles='xy', scale_units='xy', scale=1, color='b', label='Rotated')
+    plt.xlim(-1, 3)
+    plt.ylim(-1, 3)
+    plt.grid(True)
+    plt.legend()
+    plt.title('Rotation of a Vector')
+    plt.show()
+    return np, plt, v
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        **Result:** The vector is rotated by 45°.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Example 2: Scaling a Vector 0001F4A1
+        - **Original vector:** $v = [2, 1]$
+        - **Scaling matrix:**
+          $$ S = egin{bmatrix} 2 & 0 \ 0 & 0.5 nd{bmatrix} $$
+        - **Result:** Vector is stretched in $x$ and compressed in $y$
+        """
+    )
+    return
+
+
+@app.cell
+def _(np, plt, v):
+    S = np.array([[2, 0], [0, 0.5]])
+    v_scale = S @ v
+
+    plt.figure(figsize=(6,6))
+    plt.quiver(0, 0, v[0], v[1], angles='xy', scale_units='xy', scale=1, color='r', label='Original')
+    plt.quiver(0, 0, v_scale[0], v_scale[1], angles='xy', scale_units='xy', scale=1, color='g', label='Scaled')
+    plt.xlim(-1, 5)
+    plt.ylim(-1, 2)
+    plt.grid(True)
+    plt.legend()
+    plt.title('Scaling a Vector')
+    plt.show()
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        **Result:** The vector is stretched horizontally and compressed vertically.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Example 3: Shearing a Vector 0001F4A5
+        - **Original vector:** $v = [2, 1]$
+        - **Shearing matrix:**
+          $$ H = egin{bmatrix} 1 & 1.2 \ 0 & 1 nd{bmatrix} $$
+        - **Result:** Vector is slanted horizontally
+        """
+    )
+    return
+
+
+@app.cell
+def _(np, plt, v):
+    H = np.array([[1, 1.2], [0, 1]])
+    v_shear = H @ v
+
+    plt.figure(figsize=(6,6))
+    plt.quiver(0, 0, v[0], v[1], angles='xy', scale_units='xy', scale=1, color='r', label='Original')
+    plt.quiver(0, 0, v_shear[0], v_shear[1], angles='xy', scale_units='xy', scale=1, color='m', label='Sheared')
+    plt.xlim(-1, 5)
+    plt.ylim(-1, 3)
+    plt.grid(True)
+    plt.legend()
+    plt.title('Shearing a Vector')
+    plt.show()
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        **Result:** The vector is slanted horizontally.
+        """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+        ## Why are Linear Transformations Important in ML?
+        - They help manipulate and preprocess data
+        - Used in feature engineering, PCA, neural networks, and more
+        - Understanding them builds intuition for how ML algorithms work
+        """
+    )
+    return
+
+
+@app.cell
+def _():
+    import marimo as mo
+    return (mo,)
+
+
+if __name__ == "__main__":
+    app.run()

linear_algebra/linear_algebra_foundations.py

@@ -0,0 +1,185 @@
+import marimo
+
+__generated_with = "0.15.5"
+app = marimo.App()
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+    # Linear Algebra Foundations for Machine Learning
+
+    Welcome! This notebook introduces the essential concepts of linear algebra needed for machine learning, with a focus on visualization and geometric intuition.
+    """
+    )
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+    ## Topics Covered
+    - Vectors and vector operations
+    - Matrices and matrix operations
+    - Visualizing vectors and matrices
+    - Why these concepts matter in ML
+    """
+    )
+    return
+
+
+@app.cell
+def _():
+    # Import required libraries
+    import numpy as np
+    import matplotlib.pyplot as plt
+    from mpl_toolkits.mplot3d import Axes3D
+    return np, plt
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+    ## Vectors: Definition and Visualization
+    A vector is an ordered list of numbers, representing a point or direction in space.
+    """
+    )
+    return
+
+
+@app.cell
+def _(np, plt):
+    # Define two vectors in 2D
+    v1 = np.array([2, 3])
+    v2 = np.array([4, 1])
+
+    # Plot the vectors
+    plt.figure(figsize=(6,6))
+    plt.quiver(0, 0, v1[0], v1[1], angles='xy', scale_units='xy', scale=1, color='r', label='v1')
+    plt.quiver(0, 0, v2[0], v2[1], angles='xy', scale_units='xy', scale=1, color='b', label='v2')
+    plt.xlim(-1, 5)
+    plt.ylim(-1, 5)
+    plt.grid(True)
+    plt.legend()
+    plt.title('2D Vectors')
+    plt.show()
+    return v1, v2
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+    ## Vector Operations
+    Let's add two vectors and visualize the result.
+    """
+    )
+    return
+
+
+@app.cell
+def _(plt, v1, v2):
+    # Vector addition
+    v_sum = v1 + v2
+
+    plt.figure(figsize=(6,6))
+    plt.quiver(0, 0, v1[0], v1[1], angles='xy', scale_units='xy', scale=1, color='r', label='v1')
+    plt.quiver(0, 0, v2[0], v2[1], angles='xy', scale_units='xy', scale=1, color='b', label='v2')
+    plt.quiver(0, 0, v_sum[0], v_sum[1], angles='xy', scale_units='xy', scale=1, color='g', label='v1 + v2')
+    plt.xlim(-1, 7)
+    plt.ylim(-1, 7)
+    plt.grid(True)
+    plt.legend()
+    plt.title('Vector Addition')
+    plt.show()
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+    ## Matrices: Definition and Visualization
+    A matrix is a rectangular array of numbers. In ML, matrices often represent datasets or transformations.
+    """
+    )
+    return
+
+
+@app.cell
+def _(np):
+    # Define a matrix
+    A = np.array([[1, 2], [3, 4]])
+    print('Matrix A:')
+    print(A)
+    return (A,)
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+    ## Matrix-Vector Multiplication
+    Matrix multiplication can be seen as a transformation of a vector.
+    """
+    )
+    return
+
+
+@app.cell
+def _(A, v1):
+    # Apply matrix A to vector v1
+    v1_transformed = A @ v1
+    print('A @ v1 =', v1_transformed)
+    return (v1_transformed,)
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+    ## Visualizing Matrix Transformation
+    Let's see how matrix A transforms vector v1.
+    """
+    )
+    return
+
+
+@app.cell
+def _(plt, v1, v1_transformed):
+    plt.figure(figsize=(6,6))
+    plt.quiver(0, 0, v1[0], v1[1], angles='xy', scale_units='xy', scale=1, color='r', label='Original v1')
+    plt.quiver(0, 0, v1_transformed[0], v1_transformed[1], angles='xy', scale_units='xy', scale=1, color='m', label='Transformed v1')
+    plt.xlim(-1, 10)
+    plt.ylim(-1, 10)
+    plt.grid(True)
+    plt.legend()
+    plt.title('Matrix Transformation')
+    plt.show()
+    return
+
+
+@app.cell(hide_code=True)
+def _(mo):
+    mo.md(
+        r"""
+    ## Why Linear Algebra Matters in ML
+    - Data is often represented as vectors/matrices
+    - Transformations (e.g., PCA, neural networks) use matrix operations
+    - Understanding these basics helps in grasping ML algorithms
+    """
+    )
+    return
+
+
+@app.cell
+def _():
+    import marimo as mo
+    return (mo,)
+
+
+if __name__ == "__main__":
+    app.run()