University Maths Flashcards: Linear Algebra
Mathematics — Linear Algebra Flashcards
20 interactive flashcards for university-level Linear Algebra. Press Space to flip, rate 1-4
Additional Flashcard Topics
Vector Space Axioms: a vector space V over a field F satisfies closure under addition and scalar multiplication, associativity, commutativity, existence of zero vector and inverses, and distributivity. Examples: ℝⁿ, function spaces, polynomial spaces.
Linear Independence: vectors v₁,…,vₖ are linearly independent if c₁v₁ + … + cₖvₖ = 0 implies all cᵢ = 0. The dimension of a vector space is the size of any basis (maximum linearly independent set).
Rank-Nullity Theorem: for a linear map T: V → W, rank(T) + nullity(T) = dim(V). The rank is the dimension of the image; the nullity is the dimension of the kernel. This relates the “output size” to the “kernel size.”
Eigenvalues and Eigenvectors: Av = λv where v ≠ 0. The characteristic polynomial det(A - λI) = 0 gives eigenvalues. Eigenvectors span invariant subspaces where A acts as scalar multiplication.
Inner Product Spaces: a vector space with an inner product (·,·) satisfying conjugate symmetry, linearity, and positive-definiteness. The Cauchy-Schwarz inequality |⟨u,v⟩| ≤ ‖u‖‖v‖ follows from these axioms.
Spectral Theorem: a real symmetric matrix is orthogonally diagonalisable: A = QDQᵀ where Q is orthogonal and D is diagonal. This is the foundation of principal component analysis and quadratic form classification.
Intuition
Linear algebra studies vector spaces and linear maps between them. A vector space is a set closed under addition and scalar multiplication; a linear map preserves these operations. The key insight is that every linear map can be represented as a matrix, and matrix operations correspond to compositions of linear maps. Eigenvalues and eigenvectors reveal the “natural axes” along which a linear transformation directly stretches or compresses — this is the foundation of diagonalisation, SVD, and principal component analysis. Linear algebra is the language of data science, quantum mechanics, and computer graphics.
Common Pitfalls
- Dimension mismatch: The rank-nullity theorem (rank + nullity = n) applies to the domain — confusing the dimensions of domain and codomain leads to wrong nullity calculations.
- Orthogonality ≠ independence: Orthogonal vectors are always linearly independent, but linearly independent vectors are not necessarily orthogonal — don’t assume orthogonality without proof.
- Diagonalisability condition: A matrix is diagonalisable if and only if it has a complete set of eigenvectors (geometric multiplicity = algebraic multiplicity for each eigenvalue) — defective matrices cannot be diagonalised.
- Forgetting that matrix multiplication is not commutative: AB ≠ BA as a rule. This affects eigenvalue calculations: eigenvectors of AB are not necessarily eigenvectors of BA.
- Confusing null space with column space: the null space contains vectors mapped to zero; the column space contains all possible outputs. These are subspaces of different spaces (domain vs codomain).
Cross-References
- Linear Algebra: Vector spaces and linear transformations fundamentals.
- Abstract Algebra: Algebraic structures underlying vector spaces; vector spaces are modules over fields.
- Multivariable Calculus: Linear algebra in multivariable contexts; Hessians and Jacobians are matrices.
- Functional Analysis: Infinite-dimensional vector spaces extending linear algebra to function spaces.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.
Advanced Content
This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.
Derivations and Proofs
Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.
Extended Examples
Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.
Research Connections
This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.
Prerequisites
Ensure you have mastered the prerequisite material before attempting this advanced content.