Quick Reference

Section B.15 Quick Reference

Array setup and shapes. See B.1 and B.2.

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Array 🔗: np.array([...]). Read as. Create an array. Used for. All labs.
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Array conversion 🔗: np.asarray(...). Read as. Treat array-like input as an array. Used for. Helper functions.
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Reshape 🔗: .reshape(m, n). Read as. Rearrange entries into a new shape. Used for. Matrix-shaped numerical output.
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Display options 🔗: np.set_printoptions(...). Read as. Change printed display of arrays. Used for. Readability.
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Floating-point data 🔗: dtype=float. Read as. Request floating-point numerical data. Used for. SymPy-to-NumPy conversion.
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Shape 🔗: A.shape. Read as. Dimensions. Used for. Shape checks.
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Matrix products and entrywise operations. See B.4 and B.5.

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Transpose 🔗: A.T. Read as. Transpose. Used for. Residual checks and normal equations.
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Dot product 🔗: u @ v. Read as. Dot product. Used for. Cosine similarity.
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Matrix-vector product 🔗: A @ x. Read as. Matrix-vector product. Used for. Linear maps.
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Matrix product 🔗: A @ B. Read as. Matrix product. Used for. Composition and attention.
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Outer product 🔗: np.outer(g, h). Read as. Rank-one outer product. Used for. Matrix updates such as gradient steps.
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Entrywise product 🔗: u * v. Read as. Entry-by-entry product. Used for. Elementwise operations.
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Norm 🔗: np.linalg.norm(x). Read as. Norm. Used for. Distances and residuals.
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Row-wise norms 🔗: np.linalg.norm(X, axis=1). Read as. One norm per row. Used for. Row-wise cosine similarity.
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Indexing, reductions, and entrywise functions. See B.3, B.6, and B.7.

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Ranking 🔗: np.argsort(scores)[::-1]. Read as. Rank scores largest-to-smallest. Used for. Document similarity.
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First column 🔗: A[:, 0]. Read as. All rows, first column. Used for. Column extraction.
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Column block 🔗: A[:, :k]. Read as. All rows, first k columns. Used for. SVD reconstruction.
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Row block 🔗: Vt[:k, :]. Read as. First k rows, all columns. Used for. SVD reconstruction.
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Row sums 🔗: A.sum(axis=1). Read as. One sum per row. Used for. Attention normalization.
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General sum 🔗: np.sum(...). Read as. Sum selected or all entries. Used for. Totals and SVD energy.
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Cumulative energy 🔗: np.cumsum(s**2). Read as. Cumulative squared singular-value totals. Used for. SVD energy curves.
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Row maximum 🔗: np.max(..., axis=..., keepdims=True). Read as. Maximum along an axis with shape preserved. Used for. Stable softmax.
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Coordinatewise maximum 🔗: np.maximum(z, 0). Read as. Entry-by-entry maximum with zero. Used for. ReLU.
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Exponential 🔗: np.exp(x). Read as. Entry-by-entry exponential. Used for. Softmax.
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Logarithm 🔗: np.log(x). Read as. Entry-by-entry logarithm. Used for. Loss and log probabilities.
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Square root 🔗: np.sqrt(x). Read as. Entry-by-entry square root. Used for. Norms and scales.
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Hyperbolic tangent 🔗: np.tanh(x). Read as. Entry-by-entry hyperbolic tangent. Used for. Fixed hidden layer.
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Constructors and linear algebra. See B.8, B.9, and B.10.

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Ones vector 🔗: np.ones(n). Read as. Vector of ones. Used for. Test data and constant columns.
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Matching ones 🔗: np.ones_like(t). Read as. Ones with the same shape. Used for. Constant feature columns and masks.
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Matching constants 🔗: np.full_like(t, value). Read as. Fill the same shape with one value. Used for. Baseline arrays and plotting helpers.
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Zero array 🔗: np.zeros((m, n)). Read as. Make an \(m\times n\) zero array. Used for. Test data and placeholders.
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Identity matrix 🔗: np.eye(n). Read as. Make an \(n\times n\) identity matrix. Used for. Rank and inverse checks.
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Column stack 🔗: np.column_stack([...]). Read as. Build a matrix from columns. Used for. Design matrices.
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Vertical stack 🔗: np.vstack([...]). Read as. Stack arrays as rows. Used for. Point-table construction.
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Diagonal matrix 🔗: np.diag(s). Read as. Put vector entries on the diagonal. Used for. SVD reconstruction.
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Sample grid 🔗: np.linspace(a, b, N). Read as. Evenly spaced sample points. Used for. Plotting and sampling.
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Trapezoid rule 🔗: np.trapezoid(values, grid). Read as. Approximate an integral from samples. Used for. Polynomial inner products.
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Numerical rank 🔗: np.linalg.matrix_rank(A). Read as. Numerical rank. Used for. Independent directions.
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Determinant 🔗: np.linalg.det(A). Read as. Determinant of a square matrix. Used for. Invertibility checks.
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Least squares 🔗: np.linalg.lstsq(A, b, rcond=None)[0]. Read as. Least-squares coefficients. Used for. Regression, projection, and sampled polynomial approximation.
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Residual orthogonality 🔗: A.T @ r. Read as. Dot products with the columns of \(A\text{.}\) Used for. Least-squares residual checks.
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Linear solve 🔗: np.linalg.solve(A, b). Read as. Solve a square nonsingular system. Used for. Linear systems.
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General eigenvalues 🔗: np.linalg.eig(H). Read as. Eigenvalues and eigenvectors. Used for. General square matrices.
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Symmetric eigenvalues 🔗: np.linalg.eigvalsh(H). Read as. Eigenvalues of a symmetric matrix. Used for. Hessian classification.
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Symmetric eigenvectors 🔗: np.linalg.eigh(B). Read as. Symmetric eigenvalues and eigenvectors. Used for. Spectral and SVD-related computations.
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QR factorization 🔗: np.linalg.qr(A). Read as. QR factorization. Used for. Least-squares checks.
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SVD 🔗: np.linalg.svd(A, full_matrices=False). Read as. SVD. Used for. Compression.
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Polynomial approximation patterns. See B.4, B.8, B.9, and B.10.

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Sampled powers 🔗: xs**2, xs**3. Read as. Sampled powers of \(x\text{.}\) Used for. Polynomial feature columns.
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Sampled product 🔗: xx*yy. Read as. Entrywise product of sampled \(x\)- and \(y\)-coordinates. Used for. Two-variable polynomial features.
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Two-variable grid 🔗: np.meshgrid(grid, grid). Read as. Coordinate arrays for a rectangular sample grid. Used for. Two-variable sampled fits.
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Flatten grid 🔗: X.ravel(). Read as. Flatten an array into one dimension. Used for. Turning grid coordinates into sample lists.
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Fitted sampled values 🔗: A @ c. Read as. Fitted sampled values. Used for. Polynomial least-squares fits.
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Residual vector 🔗: y - A @ c. Read as. Residual vector. Used for. Residual-orthogonality checks.
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Quadratic form 🔗: h @ H @ h. Read as. Quadratic form \(\mathbf h^T H\mathbf h\text{.}\) Used for. Hessian quadratic forms.
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Python helper patterns. See B.12.

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Define function 🔗: def f(...):. Read as. Define a helper function. Used for. Repeated computations.
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Return value 🔗: return .... Read as. Output from a function. Used for. Helper functions.
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Loop 🔗: for k in range(n):. Read as. Repeat an indented block. Used for. Iterative algorithms.
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Append 🔗: losses.append(...). Read as. Add one item to a list. Used for. Recording histories.
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Plain number 🔗: float(...). Read as. Convert to a scalar. Used for. Cleaner output.
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List comprehension 🔗: [f(x) for x in values]. Read as. Build a list by looping. Used for. Compact repeated calculations.
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Dictionary comprehension 🔗: {k: v for ...}. Read as. Build a dictionary by looping. Used for. Comparing parameter choices.
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Pairing 🔗: zip(a, b). Read as. Pair entries from two iterables. Used for. Labeled loops.
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Dictionary 🔗: dict(...). Read as. Build a dictionary. Used for. Named results.
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Length 🔗: len(values). Read as. Count entries. Used for. Sizes and loops.
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Round 🔗: round(x, 3). Read as. Round a Python number. Used for. Readable output.
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Print 🔗: print(...). Read as. Display a value. Used for. Notebook output.
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Assert 🔗: assert condition. Read as. Check that a condition holds. Used for. Lab sanity checks.
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Optional attention, SymPy, checks, and plotting. See B.11, B.9, B.13, and B.14.

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Upper triangle 🔗: np.triu(..., k=1). Read as. Upper-triangular part above the diagonal. Used for. Causal mask.
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Boolean mask 🔗: .astype(bool). Read as. Convert to Boolean mask. Used for. Optional attention.
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Copy 🔗: .copy(). Read as. Make an independent copy. Used for. Optional attention.
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Mask assignment 🔗: S_masked[mask] = -1e9. Read as. Change selected entries. Used for. Optional attention masks.
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Row normalization 🔗: e / e.sum(axis=1, keepdims=True). Read as. Divide each row by its row sum. Used for. Softmax rows.
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Symbols 🔗: sp.symbols("x y"). Read as. Create symbolic variables. Used for. SymPy formulas.
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Symbolic matrix 🔗: sp.Matrix([...]). Read as. Symbolic matrix or vector. Used for. SymPy.
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Symbolic identity 🔗: sp.eye(n). Read as. SymPy identity matrix. Used for. Exact examples.
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Derivative 🔗: sp.diff(f, x). Read as. Symbolic derivative. Used for. Exact derivative checks.
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Jacobian 🔗: F.jacobian([x, y]). Read as. Symbolic Jacobian. Used for. Local linearization.
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Substitution 🔗: J.subs({...}). Read as. Substitute values. Used for. Jacobian at a point.
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NumPy conversion 🔗: np.array(..., dtype=float). Read as. Convert symbolic output to NumPy. Used for. Numerical multiplication.
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Row reduction 🔗: M.rref(). Read as. Reduced row echelon form and pivots. Used for. Exact pivot analysis.
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Nullspace 🔗: M.nullspace(). Read as. List of symbolic column-vector basis elements. Used for. Exact null spaces.
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Scalar closeness 🔗: np.isclose(a, b). Read as. Approximate scalar equality. Used for. Numerical checks.
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Approximate equality 🔗: np.allclose(x, y). Read as. Approximate equality. Used for. Numerical checks.
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Rounded display 🔗: np.round(x, 3). Read as. Round an array for display. Used for. Readable output.
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Figure setup 🔗: plt.figure(...). Read as. Start a figure. Used for. Optional plots.
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Figure and axes 🔗: plt.subplots(...). Read as. Start a figure and axes pair. Used for. Optional plots.
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Matrix image 🔗: plt.imshow(A). Read as. Display a matrix as an image. Used for. Optional attention plots.
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Color scale 🔗: plt.colorbar(...). Read as. Add a scale bar. Used for. Matrix image interpretation.
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Plot labels 🔗: plt.title(...), plt.xlabel(...), plt.ylabel(...). Read as. Add labels. Used for. Readable plots.
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Tick labels 🔗: plt.xticks(...), plt.yticks(...). Read as. Add tick labels. Used for. Token labels.
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Line plot 🔗: plt.plot(...). Read as. Draw connected points or curves. Used for. Optional geometry plots.
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Equal scales 🔗: plt.axis("equal"). Read as. Use equal horizontal and vertical scales. Used for. Optional geometry plots.
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Grid 🔗: plt.grid(True). Read as. Show a background grid. Used for. Optional geometry plots.
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Legend 🔗: plt.legend(). Read as. Show plot labels. Used for. Optional geometry plots.
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Display figure 🔗: plt.show(). Read as. Display the figure. Used for. Notebooks.
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