Linear Algebra Flashcards
Free linear algebra flashcards covering vectors, matrix operations, determinants, inverses, systems of equations, vector spaces and eigenvalues. Flip each card, mark what you know and drill what you miss.
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Vectors
| Question | Answer |
|---|---|
| Dot product of u and v? | u · v = u₁v₁ + u₂v₂ + … = |u||v| cos θ |
| When are two vectors orthogonal? | When their dot product is 0. |
| Magnitude (norm) of v = (v₁, v₂, …)? | √(v₁² + v₂² + …) |
| Unit vector in the direction of v? | v / |v| |
| Projection of u onto v? | ((u · v) / (v · v)) v |
| Cross product of two vectors in R³ gives? | A vector perpendicular to both, with magnitude |u||v| sin θ. |
| Linear combination of v₁, …, vₖ? | c₁v₁ + … + cₖvₖ |
| Span of a set of vectors? | The set of all their linear combinations. |
| Linearly independent vectors? | Vectors where the only combination equal to 0 is the one with all coefficients 0. |
Matrix operations
| Question | Answer |
|---|---|
| When can you multiply A (m × n) by B (p × q)? | When n = p. The result is m × q. |
| Is matrix multiplication commutative? | No. In general AB ≠ BA. |
| Transpose of a matrix? | Swap rows and columns: (Aᵀ)ᵢⱼ = Aⱼᵢ. |
| (AB)ᵀ = ? | BᵀAᵀ |
| Identity matrix I? | A square matrix with 1s on the diagonal and 0s elsewhere, so AI = IA = A. |
| Inverse of A? | The matrix A⁻¹ with AA⁻¹ = A⁻¹A = I. |
| (AB)⁻¹ = ? | B⁻¹A⁻¹ |
| When does a square matrix have an inverse? | When its determinant is not zero. |
| Inverse of a 2 × 2 matrix [[a, b], [c, d]]? | (1 / (ad − bc)) [[d, −b], [−c, a]] |
Determinants
| Question | Answer |
|---|---|
| Determinant of [[a, b], [c, d]]? | ad − bc |
| What does det A = 0 mean? | A is singular (not invertible), and its rows are linearly dependent. |
| det(AB) = ? | det(A) det(B) |
| det(Aᵀ) = ? | det(A) |
| det(kA) for an n × n matrix? | kⁿ det(A) |
| What happens to the determinant if you swap two rows? | It changes sign. |
| What does | det A| measure geometrically?|The factor by which the transformation scales area (2D) or volume (3D). |
Systems and vector spaces
| Question | Answer |
|---|---|
| What is Gaussian elimination? | Using row operations to reduce an augmented matrix to row echelon form, then back substitute. |
| Three row operations? | Swap two rows, multiply a row by a nonzero constant, add a multiple of one row to another. |
| A system Ax = b has a unique solution when? | A is square and invertible (det A ≠ 0). |
| Rank of a matrix? | The number of linearly independent rows (or columns). |
| Rank-nullity theorem? | rank(A) + nullity(A) = number of columns of A. |
| Null space of A? | The set of all x with Ax = 0. |
| Basis of a vector space? | A linearly independent set that spans the space. |
| Dimension of a vector space? | The number of vectors in any basis. |
Eigenvalues
| Question | Answer |
|---|---|
| Definition of an eigenvector v and eigenvalue λ? | Av = λv with v ≠ 0. |
| How do you find eigenvalues? | Solve det(A − λI) = 0 (the characteristic equation). |
| Sum of the eigenvalues of A? | The trace of A (sum of the diagonal entries). |
| Product of the eigenvalues of A? | The determinant of A. |
| When is a matrix diagonalizable? | When it has n linearly independent eigenvectors. |
| Eigenvalues of a triangular matrix? | The diagonal entries. |
| Eigenvalues of a symmetric real matrix? | All real, with orthogonal eigenvectors. |
Frequently asked questions
What are the main ideas in linear algebra?
Vectors, linear transformations, matrices, solving linear systems, vector spaces and eigenvalues are the core ideas of a first course.
Do I need calculus for linear algebra?
Usually no. A first course needs algebra and basic vector ideas, though many courses are taught alongside calculus.
Sources
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