Linear Algebra Fundamentals

Describe what a system of two linear equations represents graphically (intersecting, parallel, or coincident lines).

Explain the meaning of a system having no solution, one solution, or infinitely many solutions.

Demonstrate how substitution finds the intersection point of two lines for a 2x2 system.

Define a vector as an ordered list of numbers and distinguish it from a scalar.

Re-interpret a system of linear equations as a question about linear combinations of vectors.

Represent vectors geometrically as arrows in two and three dimensions.

Perform vector addition and subtraction, and describe their geometric interpretation (head-to-tail rule).

Perform scalar multiplication on a vector and describe its geometric effect (scaling).

Relate the Pythagorean theorem to the formula for a vector's length (norm).

Compute the length (norm) of a vector and interpret it as the vector's magnitude.

Define a unit vector and explain the process of normalizing a vector.

Compute the dot product of two vectors.

Interpret the dot product geometrically in terms of the angle between vectors.

Explain how the dot product measures similarity or alignment between two vectors.

Use the dot product to determine if two vectors are orthogonal (perpendicular).

Describe vector projection intuitively as finding the 'shadow' of one vector onto another.

Explain how the dot product is used to compute vector projections.

Define a matrix and describe its dimensions (rows and columns).

Describe the process of multiplying a matrix by a vector.

Interpret matrix-vector multiplication as a linear transformation of space.

Identify the matrices for 2D scaling, reflection, and rotation, and interpret their geometric effects on vectors.

Explain how applying a matrix to all points in a shape transforms that shape.

Perform matrix addition and scalar multiplication, and explain their element-wise nature.

Describe the transpose of a matrix as swapping its rows and columns.

Compute the product of two matrices.

Explain how composing two transformations corresponds to multiplying their matrices.

Represent a system of linear equations in the matrix form Ax = b.

Interpret the matrix equation Ax = b as transforming a vector x into a vector b.

Compute the determinant of a 2x2 matrix.

Explain the geometric meaning of a 2D determinant as the factor by which a transformation scales area.

Explain the relationship between a zero determinant and a transformation that collapses space into a lower dimension.

Define the inverse of a matrix conceptually as a transformation that reverses the original.

Connect the value of the determinant to whether a matrix is invertible.

Describe the span of a set of vectors as all points reachable through their linear combinations.

Interpret the column space of a matrix as the span of its column vectors.

Explain the column space geometrically as the set of all possible outputs of a matrix transformation.

Connect the column space to the solvability of Ax = b, explaining that a solution exists if and only if b is in the column space of A.

Define linear independence intuitively as a set of vectors where none are redundant.

Explain the concept of a basis as a minimal set of vectors that spans a space.

Define the rank of a matrix as the dimension of its column space.

Relate rank to the number of dimensions in the output of a transformation.

Describe the null space of a matrix as the set of all input vectors that are squashed to the zero vector.

Connect the null space to the uniqueness of solutions for Ax = b.

Define eigenvalues and eigenvectors conceptually as the scaling factors and invariant directions of a linear transformation.

Interpret eigenvectors geometrically as vectors that are only stretched or shrunk by a transformation, not rotated.

Explain what eigenvalues reveal about how a transformation stretches, compresses, or flips space along eigenvector directions.

Describe how an eigenbasis can simplify the understanding of a linear transformation.

Summarize the story of Ax = b by relating the column space, null space, and rank to the existence and uniqueness of solutions.

Create a concept map linking invertibility, rank, the determinant, and the null space of a square matrix.

Contrast the effect of a transformation on an eigenvector versus an arbitrary vector.

Explain how the core ideas of linear algebra (span, basis, transformations) provide a language to describe high-dimensional spaces.

Explain how data points are represented as vectors and datasets as matrices in AI.

Describe how dot products are used to measure similarity between data representations (e.g., for recommendation systems).

Explain the role of matrix multiplication in computing a neural network layer's output.

Describe the intuition behind Principal Component Analysis (PCA) as finding a new basis of eigenvectors to reduce dimensionality.

Explain the concept of embeddings as meaningful vector representations of data in a high-dimensional space.

Describe at a high level how matrix operations underpin the attention mechanism in transformers.