--- title: Multiple Linear Regression: From One Feature to a Hyperplane description: An educational walkthrough explaining Multiple Linear Regression as an extension of Simple Linear Regression, covering the geometric intuition of lines,... canonical: https://daily.dev/posts/multiple-linear-regression-from-one-feature-to-a-hyperplane-ruyegs6qm twitter:card: summary_large_image twitter:site: @dailydotdev og:type: website og:site_name: daily.dev og:title: Multiple Linear Regression: From One Feature to a Hyperplane | daily.dev og:description: An educational walkthrough explaining Multiple Linear Regression as an extension of Simple Linear Regression, covering the geometric intuition of lines,... og:url: https://daily.dev/posts/multiple-linear-regression-from-one-feature-to-a-hyperplane-ruyegs6qm og:image: https://api.daily.dev/og/posts/RuYEGs6QM.png og:image:alt: Multiple Linear Regression: From One Feature to a Hyperplane og:image:width: 1200 og:image:height: 630 og:locale: en --- > ## Documentation Index > Fetch the complete documentation index at: https://daily.dev/llms.txt > Use this file to discover all available pages before exploring further. # Multiple Linear Regression: From One Feature to a Hyperplane **[Medium](https://daily.dev/sources/medium_js)** · 12 min read · 0 upvotes · 0 comments ## Summary An educational walkthrough explaining Multiple Linear Regression as an extension of Simple Linear Regression, covering the geometric intuition of lines, planes, and hyperplanes, the matrix formulation of the model, derivation of the Normal Equation via Ordinary Least Squares, and dimension analysis of the matrices involved. It then implements the model using Scikit-Learn, evaluates it with MAE/MSE/RMSE/R², builds a from-scratch implementation using NumPy's Normal Equation, compares OLS against Gradient Descent, and discusses pitfalls like multicollinearity and overfitting when adding features. ## Full article daily.dev links to this article rather than hosting it. Read it at the original source: ## Questions this post answers ### How do you derive the coefficients for multiple linear regression using the normal equation? Coefficients are found by minimizing the sum of squared residuals, which leads to the closed-form Normal Equation beta = (X^T X)^-1 X^T y, where X is the feature matrix (including a column of ones for the intercept) and y is the target vector. This differs from gradient descent, which iteratively updates parameters instead of solving analytically. _daily.dev surfaces regression math explainers for anyone implementing normal equation solutions from scratch._ ### What is the difference between using the normal equation and gradient descent for linear regression? The normal equation computes an exact analytical solution by directly solving beta = (X^T X)^-1 X^T y, while gradient descent starts with initial parameter guesses and iteratively moves toward a better solution by minimizing loss. The normal equation works well for smaller problems, but computing a matrix inverse gets expensive as feature count grows, making gradient descent preferable for large datasets or models without a closed-form solution. _Comparing optimization approaches like these is easier when developers track ML technique trade-offs on daily.dev._ ### Why shouldn't you compare coefficient magnitudes to determine feature importance in multiple linear regression? Coefficient magnitude depends heavily on the scale of the features, so a feature measured in thousands can have a much smaller coefficient than one measured between 0 and 1 while still having substantial influence on the prediction. Comparing raw coefficients across differently scaled features can therefore mislead conclusions about which feature matters most. _Developers refining model interpretation pitfalls like this often keep up via daily.dev._ ## Similar posts on daily.dev - [Multiple Linear Regression Explained Simply \(Part 1\)](https://daily.dev/posts/multiple-linear-regression-explained-simply-part-1--05gzjtxbz) · Towards Data Science · 0 upvotes · 0 comments - [Linear Regression For Machine Learning….](https://daily.dev/posts/linear-regression-for-machine-learning--cbrwzucbu) · Medium · 0 upvotes · 0 comments - [Linear Regression Is Actually a Projection Problem \(Part 2: From Projections to Predictions\)](https://daily.dev/posts/linear-regression-is-actually-a-projection-problem-part-2-from-projections-to-predictions--wejmyibaf) · Towards Data Science · 0 upvotes · 0 comments - [Linear Regression: The Simplest Model That Actually Works](https://daily.dev/posts/linear-regression-the-simplest-model-that-actually-works-gh0lpkiuu) · Medium · 0 upvotes · 0 comments --- Tags: [#machine-learning](https://daily.dev/tags/machine-learning), [#python](https://daily.dev/tags/python), [#math](https://daily.dev/tags/math), [#scikit](https://daily.dev/tags/scikit), [#linear-regression](https://daily.dev/tags/linear-regression) [View this post on daily.dev](https://daily.dev/posts/multiple-linear-regression-from-one-feature-to-a-hyperplane-ruyegs6qm) ```json {"@context":"https://schema.org","@graph":[{"@type":"Organization","@id":"https://daily.dev/#organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180},"sameAs":["https://twitter.com/dailydotdev","https://github.com/dailydotdev","https://www.linkedin.com/company/daily-dev-ltd"]},{"@type":"WebSite","@id":"https://daily.dev/#website","url":"https://daily.dev","name":"daily.dev","publisher":{"@id":"https://daily.dev/#organization"},"potentialAction":{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"https://daily.dev/search?q={search_term_string}"},"query-input":"required name=search_term_string"}}]} {"@context":"https://schema.org","@type":"TechArticle","headline":"Multiple Linear Regression: From One Feature to a Hyperplane","url":"https://daily.dev/posts/multiple-linear-regression-from-one-feature-to-a-hyperplane-ruyegs6qm","mainEntityOfPage":{"@type":"WebPage","@id":"https://daily.dev/posts/multiple-linear-regression-from-one-feature-to-a-hyperplane-ruyegs6qm"},"datePublished":"2026-08-18T06:02:12.412Z","dateModified":"2026-09-13T18:31:41.099Z","description":"An educational walkthrough explaining Multiple Linear Regression as an extension of Simple Linear Regression, covering the geometric intuition of lines,...","image":"https://media.daily.dev/image/upload/f_auto,q_auto/v1/posts/5a3667aa16d717dea54c78bfd96e8083?_a=AQAEuop","thumbnailUrl":"https://media.daily.dev/image/upload/f_auto,q_auto/v1/posts/5a3667aa16d717dea54c78bfd96e8083?_a=AQAEuop","isAccessibleForFree":true,"articleSection":"Medium","inLanguage":"en","publisher":{"@type":"Organization","name":"daily.dev","url":"https://daily.dev","logo":{"@type":"ImageObject","url":"https://daily.dev/apple-touch-icon.png","width":180,"height":180}},"author":{"@type":"Organization","name":"Medium","logo":"https://media.daily.dev/image/upload/t_logo,f_auto/v1/logos/medium","url":"https://daily.dev/sources/medium_js"},"commentCount":0,"discussionUrl":"https://daily.dev/posts/multiple-linear-regression-from-one-feature-to-a-hyperplane-ruyegs6qm","interactionStatistic":[{"@type":"InteractionCounter","interactionType":{"@type":"LikeAction"},"userInteractionCount":0},{"@type":"InteractionCounter","interactionType":{"@type":"CommentAction"},"userInteractionCount":0}],"keywords":"machine-learning,python,math,scikit,linear-regression","timeRequired":"PT12M"} {"@context":"https://schema.org","@type":"BreadcrumbList","itemListElement":[{"@type":"ListItem","position":1,"name":"Home","item":"https://daily.dev"},{"@type":"ListItem","position":2,"name":"Medium","item":"https://daily.dev/sources/medium_js"},{"@type":"ListItem","position":3,"name":"Multiple Linear Regression: From One Feature to a Hyperplane"}]} {"@context":"https://schema.org","@type":"FAQPage","@id":"https://daily.dev/posts/multiple-linear-regression-from-one-feature-to-a-hyperplane-ruyegs6qm#faq","mainEntity":[{"@type":"Question","name":"How do you derive the coefficients for multiple linear regression using the normal equation?","acceptedAnswer":{"@type":"Answer","text":"Coefficients are found by minimizing the sum of squared residuals, which leads to the closed-form Normal Equation beta = (X^T X)^-1 X^T y, where X is the feature matrix (including a column of ones for the intercept) and y is the target vector. 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