All atlases/Linear Algebra

Mathematics Atlas

Linear Algebra

A rigorous, matrix-first introduction to linear algebra that moves from solving systems of equations to the spectral theorem and singular-value decomposition, with emphasis on both computation and conceptual understanding. The course follows the narrative arc of Gilbert Strang’s Introduction to Linear Algebra, giving students the tools to work with high-dimensional data, solve large linear systems, and understand the geometry of linear transformations that underpin modern machine-learning and engineering models.

10 concepts20 checks30 cards

Next: Vectors, Linear Combinations and Span in ℝⁿ

Atlas map

Vectors, Linear Combinations and the Geometry of Linear Equations

Recommended next1. Vectors, Linear Combinations and Span in ℝⁿ

← Swipe to explore the full concept map →

current pathprerequisite
Vectors, Linear Combinations and Span in ℝⁿ appears earlier in the syllabus and supports Gaussian Elimination and Echelon Form.Gaussian Elimination and Echelon Form appears earlier in the syllabus and supports The Column Space, Nullspace, Row Space and Left Nullspace.The Column Space, Nullspace, Row Space and Left Nullspace appears earlier in the syllabus and supports Linear Independence and Basis.Linear Independence and Basis appears earlier in the syllabus and supports Orthogonal Projections and the Least-Squares Problem.Orthogonal Projections and the Least-Squares Problem appears earlier in the syllabus and supports Definition and Properties of the Determinant.Definition and Properties of the Determinant appears earlier in the syllabus and supports Eigenvalues and Eigenvectors: Definition and Computation.Eigenvalues and Eigenvectors: Definition and Computation appears earlier in the syllabus and supports Real Symmetric Matrices and the Spectral Theorem.Real Symmetric Matrices and the Spectral Theorem appears earlier in the syllabus and supports Low-Rank Approximation and the Eckart–Young Theorem.Low-Rank Approximation and the Eckart–Young Theorem appears earlier in the syllabus and supports Applications: Graphs, Networks and Markov Chains.prerequisite

prerequisite relationship

Vectors, Linear Combinations and Span in ℝⁿ -> Gaussian Elimination and Echelon Form

Vectors, Linear Combinations and Span in ℝⁿ appears earlier in the syllabus and supports Gaussian Elimination and Echelon Form.

ready

Recommended next

Vectors, Linear Combinations and Span in ℝⁿ

We introduce vectors as arrows and as columns of numbers, define vector addition and scalar multiplication geometrically and algebraically, and build linear combinations to reach the concept of span.

Step 1 / 5

Vectors, Linear Combinations and Span in ℝⁿ: the core idea

We introduce vectors as arrows and as columns of numbers, define vector addition and scalar multiplication geometrically and algebraically, and build linear combinations to reach the concept of span. The key thing to notice is: vector as column of numbers vs. arrow in ℝⁿ. A useful example is RGB color space: any color is a linear combination of (255,0,0), (0,255,0), (0,0,255); the span is every color on the screen.. Do not treat this as a vocabulary item; the point is to use it to reason about a new situation.

Where would Vectors, Linear Combinations and Span in ℝⁿ show up in an everyday decision or news headline?

Look for the hidden relationship in the example: RGB color space: any color is a linear combination of (255,0,0), (0,255,0), (0,0,255); the span is every color on the screen..

Vectors, Linear Combinations and Span in ℝⁿ: the core idea