Linear Algebra Practice (Interactive)

Intuition

The mathematics of vectors and transformations: Linear algebra is the mathematics of flat things — vectors, lines, planes, and the transformations that map them to each other. It is the language of computer graphics, quantum mechanics, and machine learning.

Why it matters: Every data point can be viewed as a vector, every operation as a matrix, and every model as a linear transformation. Understanding linear algebra is essential for machine learning, graphics, and scientific computing.

The key insight: Eigenvalues and eigenvectors reveal the “natural directions” of a transformation — the directions that only get stretched, not rotated. This is why PCA (principal component analysis) finds the most important directions in data.

University Mathematics — Linear Algebra Practice

10 auto-graded practice problems at medium to hard difficulty. Select an answer, submit, and review the explanation.

Matrices

Vector Spaces

Eigenvalues and Eigenvectors

Linear Transformations

Inner Products

Common Mistakes

Forgetting that matrix multiplication is not commutative: $AB \neq BA$ as a rule. The $(1,2)$ entry of $AB$ is computed from row 1 of $A$ and column 2 of $B$, which is entirely different from row 1 of $B$ and column 2 of $A$. Always compute products in the correct order and never assume $AB = BA$.

Confusing linear independence with linear dependence criteria: Having no vector as a scalar multiple of another is necessary but not sufficient for linear independence in dimensions $\geq 3$. For example, $(1,2,3)$, $(2,5,7)$, and $(1,3,4)$ have no pair of scalar multiples but are still linearly dependent because $v_3 = v_2 - v_1$. Always check whether one vector is a general linear combination of the others.

Mixing up the rank-nullity theorem with dimension counting: The rank-nullity theorem states $\text{rank}(T) + \text{nullity}(T) = \dim(\text{domain})$, not $\dim(\text{codomain})$. For $T: \mathbb{R}^3 \to \mathbb{R}^3$ with rank 2, the nullity is $3 - 2 = 1$, but for $T: \mathbb{R}^3 \to \mathbb{R}^5$ with rank 2, the nullity is still $3 - 2 = 1$ because the domain determines the constraint.

Cross-References

• Site Home: Main landing page for Mathematics notes.
• Linear Algebra: Vector spaces, matrices, and linear transformations.
• Real Analysis: Rigorous treatment of real numbers and calculus.
• Practice: Practice problems for revision.

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.