Linear Algebra - Matrices, Vectors & Transformations Flashcards

Question 1

Null Space (Kernel)

Accepted Answer

The set of all vectors $x$ that satisfy the homogeneous system $A x = 0$, denoted $\	ext{Null}(A)$ or $\ker(A)$. Its dimension is called the **nullity**.

Question 2

Vector Space

Accepted Answer

A set $V$ equipped with vector addition and scalar multiplication satisfying eight fundamental axioms, including closure under both operations, associativity, commutativity, identity elements, and distributivity.

Question 3

Linear Independence

Accepted Answer

A set of vectors $\{v_1, v_2, \dots, v_n\}$ is linearly independent if the equation $c_1 v_1 + c_2 v_2 + \dots + c_n v_n = 0$ holds only when all scalars $c_1 = c_2 = \dots = c_n = 0$ (the trivial solution).

Question 4

Span

Accepted Answer

The set of all possible linear combinations of a collection of vectors $\{v_1, v_2, \dots, v_n\}$, denoted as $\	ext{span}(v_1, \dots, v_n)$. It forms a subspace of the vector space.

Question 5

Basis and Dimension

Accepted Answer

A **basis** for a vector space $V$ is a set of linearly independent vectors that spans $V$. The **dimension** $\dim(V)$ is the total number of vectors in any basis of $V$.

Question 6

Dot Product (Algebraic & Geometric)

Accepted Answer

For vectors $u, v \in \mathbb{R}^n$, the dot product is $u \cdot v = \sum_{i=1}^n u_i v_i = \|u\| \|v\| \cos(\	heta)$, where $\	heta$ is the angle between them. If $u \cdot v = 0$, the vectors are orthogonal.

Question 7

Cross Product

Accepted Answer

For vectors $u, v \in \mathbb{R}^3$, $u \	imes v$ yields a vector orthogonal to both $u$ and $v$ with magnitude $\|u \	imes v\| = \|u\| \|v\| \sin(\	heta)$, which equals the area of the parallelogram formed by $u$ and $v$.

Question 8

Linear Transformation

Accepted Answer

A mapping $T: V \	o W$ satisfying two rules for all $u, v \in V$ and scalars $c$:
1. Additivity: $T(u + v) = T(u) + T(v)$
2. Homogeneity: $T(c u) = c T(u)$

Question 9

Matrix Rank

Accepted Answer

The maximum number of linearly independent column (or row) vectors in a matrix $A$, denoted $\	ext{rank}(A)$. It measures the dimension of the column space $\	ext{Col}(A)$ (i.e., the dimension of the output space).

Question 10

Rank-Nullity Theorem

Accepted Answer

For an $m \	imes n$ matrix $A$, $\	ext{rank}(A) + \	ext{nullity}(A) = n$, where $n$ is the total number of columns in $A$ (the dimension of the input space).

Question 11

Geometric Meaning of Determinant

Accepted Answer

The determinant $\det(A)$ represents the factor by which a transformation scales areas (in 2D) or volumes (in higher dimensions). A negative determinant indicates an orientation flip, while $\det(A) = 0$ means space is collapsed into a lower dimension.

Question 12

Eigenvalue and Eigenvector

Accepted Answer

A non-zero vector $v$ is an **eigenvector** of square matrix $A$ with **eigenvalue** $\lambda$ if $A v = \lambda v$. The transformation stretches or compresses $v$ by factor $\lambda$ without changing its line of orientation.

Question 13

Calculating Eigenvalues

Accepted Answer

Eigenvalues are found by solving the **characteristic equation** $\det(A - \lambda I) = 0$, where $I$ is the identity matrix.

Question 14

Calculating Eigenvectors

Accepted Answer

Once an eigenvalue $\lambda$ is found, substitute it back into $(A - \lambda I) v = 0$ and solve the non-trivial system for vector $v$ to obtain the corresponding eigenvectors.

Question 15

Eigenspace

Accepted Answer

The set of all eigenvectors corresponding to a specific eigenvalue $\lambda$, together with the zero vector. It is equal to the kernel/null space $\	ext{Null}(A - \lambda I)$.

Question 16

Matrix Diagonalization

Accepted Answer

A matrix $A$ is diagonalizable if $A = P D P^{-1}$, where $D$ is a diagonal matrix containing the eigenvalues of $A$, and $P$ is a matrix whose columns are the linearly independent eigenvectors of $A$.

Question 17

Change of Basis Matrix

Accepted Answer

A matrix $P_{\mathcal{C} \leftarrow \mathcal{B}}$ that translates coordinates from basis $\mathcal{B}$ to basis $\mathcal{C}$ such that $[x]_{\mathcal{C}} = P_{\mathcal{C} \leftarrow \mathcal{B}} [x]_{\mathcal{B}}$.

Question 18

Orthogonal Matrix

Accepted Answer

A square matrix $Q$ whose columns and rows are orthonormal vectors. It satisfies $Q^T Q = Q Q^T = I$, which implies $Q^{-1} = Q^T$.

Question 19

Gram-Schmidt Process

Accepted Answer

An algorithm for constructing an orthogonal (or orthonormal) basis from a set of linearly independent vectors $\{v_1, \dots, v_k\}$ by sequentially subtracting orthogonal projections.

Question 20

Orthogonal Projection

Accepted Answer

The closest vector in a subspace $W$ to a given vector $v$. Given an orthonormal basis $\{u_1, \dots, u_k\}$ for $W$, $\	ext{proj}_W(v) = \sum_{i=1}^k (v \cdot u_i) u_i$.

Question 21

Spectral Theorem for Symmetric Matrices

Accepted Answer

Any real symmetric matrix ($A = A^T$) has only real eigenvalues and can be orthogonally diagonalized as $A = Q D Q^T$, where $Q$ is an orthogonal matrix of eigenvectors.

Linear Algebra - Matrices, Vectors & Transformations

Cards in this deck

Related Flashcard Sets

JLPT N4 - Grammar Patterns & Kanji

German B1 - Intermediate Grammar & Vocabulary

Kubernetes Fundamentals & Orchestration

CFA Level I - Ethical and Professional Standards