Linear Algebra
77 flashcards by MentisLibrary
Vectors, matrices, determinants, eigenvalues, and the four fundamental subspaces, with the geometry behind each idea made explicit. The mathematics that powers graphics, data science, and machine learning.
- What is a vector?
- An ordered list of numbers, which can represent a magnitude and direction or a point in space
- What is a linear combination of vectors?
- A sum of scalar multiples: $c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \cdots + c_k\mathbf{v}_k$
- What is the span of a set of vectors?
- The set of all their linear combinations
- What does it mean for vectors to be linearly independent?
- The only linear combination equal to zero is the one with all coefficients zero
- What is a subspace?
- A subset containing the zero vector and closed under addition and scalar multiplication
- What is a basis?
- A linearly independent set that spans the space
- What is the dimension of a vector space?
- The number of vectors in any basis
- How do you multiply two matrices?
- Entry $(i,j)$ of the product is the dot product of row $i$ of $A$ with column $j$ of $B$
- What is the identity matrix?
- The square matrix with ones on the diagonal and zeros elsewhere, satisfying $AI = IA = A$
- What is the transpose of a matrix?
- The matrix with rows and columns swapped, so $(A^T)_{ij} = A_{ji}$
- What is a symmetric matrix?
- One equal to its own transpose, $A = A^T$
- What is the inverse of a matrix?
- The matrix $A^{-1}$ with $AA^{-1} = A^{-1}A = I$
- When is a matrix invertible?
- Exactly when its determinant is nonzero, equivalently full rank, equivalently trivial null space
- What is Gaussian elimination?
- Using row operations to reduce a matrix to row echelon form so the system can be back substituted
- What is reduced row echelon form?
- Echelon form where each pivot is 1 and is the only nonzero entry in its column
- What is a pivot?
- The leading nonzero entry of a row in echelon form, whose column is a pivot column
- When does a linear system have no solution, one solution, or infinitely many?
- No solution if a row reduces to $[0\ \cdots\ 0 \mid c]$ with $c \neq 0$, one solution if every variable is basic, infinitely many if a free variable exists
- What is the null space of a matrix?
- The set of all $\mathbf{x}$ with $A\mathbf{x} = \mathbf{0}$
- What is the column space of a matrix?
- The span of its columns, which is the set of all achievable outputs $A\mathbf{x}$
- What is the rank of a matrix?
- The number of pivots, equal to the dimension of the column space and of the row space
- What is the rank-nullity theorem?
- $\text{rank}(A) + \text{nullity}(A) = n$, the number of columns
- What does the determinant of a $2\times2$ matrix measure?
- $ad-bc$, the signed area scaling factor of the transformation
- How do you compute a $3\times3$ determinant by cofactor expansion?
- Expand along a row or column, alternating signs and multiplying by $2\times2$ minors
- What are the key determinant properties?
- $\det(AB)=\det A\det B$, $\det(A^T)=\det A$, and $\det(A^{-1})=\dfrac{1}{\det A}$
- What does a zero determinant tell you?
- The matrix is singular, its columns are dependent, and it collapses space to a lower dimension
- What is an eigenvector and an eigenvalue?
- A nonzero vector $\mathbf{v}$ with $A\mathbf{v} = \lambda\mathbf{v}$, and its scaling factor $\lambda$
- How do you find eigenvalues?
- Solve the characteristic equation $\det(A-\lambda I) = 0$
- How do you find the eigenvectors for a given eigenvalue?
- Solve $(A-\lambda I)\mathbf{x} = \mathbf{0}$, whose null space is the eigenspace
- What is the relationship between eigenvalues, trace, and determinant?
- The eigenvalues sum to the trace and multiply to the determinant
- When is a matrix diagonalizable?
- When it has $n$ linearly independent eigenvectors, giving $A = PDP^{-1}$