MIT 18.06 - Linear Algebra: Vectors, Matrices & Eigenvalues Flashcards

Question 1

Dot Product (Scalar Product)

Accepted Answer

An operation taking two equal-length sequences of numbers and returning a single scalar: $\mathbf{u} \cdot \mathbf{v} = \sum_{i=1}^n u_i v_i = \|\mathbf{u}\| \|\mathbf{v}\| \cos(	heta)$. Two non-zero vectors are **orthogonal** if their dot product is $0$.

Question 2

Cross Product

Accepted Answer

A binary operation on two vectors in 3D space $\mathbb{R}^3$ that yields a vector perpendicular to both, with magnitude equal to the area of the parallelogram spanned by them: $\mathbf{u} 	imes \mathbf{v} = \|\mathbf{u}\| \|\mathbf{v}\| \sin(	heta) \mathbf{n}$.

Question 3

Matrix Multiplication & Non-Commutativity

Accepted Answer

For matrices $A$ ($m 	imes n$) and $B$ ($n 	imes p$), the entry $(AB)_{ij}$ is the dot product of row $i$ of $A$ and column $j$ of $B$. Matrix multiplication is associative $((AB)C = A(BC))$ and distributive, but **not commutative** ($AB 
eq BA$ in general).

Question 4

Matrix Inverse ($A^{-1}$)

Accepted Answer

A square matrix $A$ is invertible (non-singular) if there exists a matrix $A^{-1}$ such that $A A^{-1} = A^{-1} A = I$. $A$ is invertible if and only if $\det(A) 
eq 0$ and $A$ has full rank.

Question 5

Gaussian Elimination & RREF

Accepted Answer

An algorithm using elementary row operations (swapping rows, scaling rows, adding row multiples) to transform an augmented matrix into **Row Reduced Echelon Form (RREF)** to solve systems $A\mathbf{x} = \mathbf{b}$ and find matrix rank.

Question 6

Column Space $C(A)$

Accepted Answer

The set of all linear combinations of the columns of matrix $A$ (also called the range of $A$). $A\mathbf{x} = \mathbf{b}$ has a solution if and only if vector $\mathbf{b}$ lies in $C(A)$.

Question 7

Null Space $N(A)$

Accepted Answer

The set of all solution vectors $\mathbf{x}$ to the homogeneous equation $A\mathbf{x} = \mathbf{0}$. $N(A)$ is a subspace of $\mathbb{R}^n$ for an $m 	imes n$ matrix $A$.

Question 8

Four Fundamental Subspaces

Accepted Answer

For an $m 	imes n$ matrix $A$ with rank $r$:
1. Column Space $C(A)$ in $\mathbb{R}^m$ (dim $r$)
2. Null Space $N(A)$ in $\mathbb{R}^n$ (dim $n - r$)
3. Row Space $C(A^T)$ in $\mathbb{R}^n$ (dim $r$)
4. Left Null Space $N(A^T)$ in $\mathbb{R}^m$ (dim $m - r$)

Question 9

Rank-Nullity Theorem

Accepted Answer

For any matrix $A$ with $n$ columns, the sum of its rank and nullity equals $n$:
$$	ext{rank}(A) + 	ext{nullity}(A) = n$$
where $	ext{rank}(A) = \dim(C(A))$ and $	ext{nullity}(A) = \dim(N(A))$.

Question 10

Determinant Properties

Accepted Answer

A scalar metric for square matrices where:
- $\det(I) = 1$
- $\det(AB) = \det(A)\det(B)$
- $\det(A^T) = \det(A)$
- $\det(A^{-1}) = \frac{1}{\det(A)}$
- Swapping two rows flips the sign of $\det(A)$.

Question 11

Eigenvalues and Eigenvectors

Accepted Answer

For a square matrix $A$, a non-zero vector $\mathbf{x}$ is an **eigenvector** if multiplying it by $A$ only scales it by a factor $\lambda$ (**eigenvalue**):
$$A\mathbf{x} = \lambda \mathbf{x}$$

Question 12

Characteristic Equation

Accepted Answer

The equation used to find the eigenvalues $\lambda$ of a square matrix $A$:
$$\det(A - \lambda I) = 0$$
The roots of this characteristic polynomial yield the matrix eigenvalues.

Question 13

Trace and Determinant via Eigenvalues

Accepted Answer

For an $n 	imes n$ matrix $A$ with eigenvalues $\lambda_1, \dots, \lambda_n$:
- **Trace**: $	ext{Tr}(A) = \sum_{i=1}^n a_{ii} = \sum_{i=1}^n \lambda_i$
- **Determinant**: $\det(A) = \prod_{i=1}^n \lambda_i$

Question 14

Matrix Diagonalization

Accepted Answer

A square matrix $A$ can be factorized as $A = S \Lambda S^{-1}$ if $A$ has $n$ linearly independent eigenvectors. Here, $S$ contains the eigenvectors as columns, and $\Lambda$ is a diagonal matrix of eigenvalues.

Question 15

Spectral Theorem (Symmetric Matrices)

Accepted Answer

If $A$ is a real symmetric matrix ($A = A^T$), then:
1. All eigenvalues of $A$ are real.
2. Eigenvectors corresponding to distinct eigenvalues are orthogonal.
3. $A$ can be factorized as $A = Q \Lambda Q^T$, where $Q$ is an orthogonal matrix.

Question 16

Orthogonal Matrices

Accepted Answer

A square matrix $Q$ whose columns and rows are orthonormal vectors. It satisfies $Q^T Q = Q Q^T = I$, meaning $Q^{-1} = Q^T$. Orthogonal transformations preserve lengths and angles.

Question 17

Projection Matrix & Least Squares

Accepted Answer

The matrix $P$ that projects a vector onto the column space $C(A)$ is $P = A(A^T A)^{-1} A^T$. In least-squares approximation ($A\mathbf{x} \approx \mathbf{b}$), the optimal solution satisfies $A^T A \hat{\mathbf{x}} = A^T \mathbf{b}$.

Question 18

Linear Transformations

Accepted Answer

A mapping $T: V 	o W$ satisfying $T(\mathbf{u} + \mathbf{v}) = T(\mathbf{u}) + T(\mathbf{v})$ and $T(c\mathbf{u}) = cT(\mathbf{u})$. Every linear transformation between finite-dimensional spaces can be represented as a matrix multiplication $T(\mathbf{x}) = A\mathbf{x}$.

Question 19

Positive Definite Matrices

Accepted Answer

A symmetric matrix $A$ is positive definite if $\mathbf{x}^T A \mathbf{x} > 0$ for all non-zero vectors $\mathbf{x}$. Equivalently, all eigenvalues are strictly positive ($\lambda_i > 0$) and all upper-left subdeterminants (minors) are positive.

Question 20

Singular Value Decomposition (SVD)

Accepted Answer

A fundamental matrix factorization applicable to any $m 	imes n$ matrix $A$:
$$A = U \Sigma V^T$$
where $U$ ($m 	imes m$) and $V$ ($n 	imes n$) are orthogonal matrices, and $\Sigma$ ($m 	imes n$) contains non-negative singular values $\sigma_i = \sqrt{\lambda_i(A^T A)}$.

MIT 18.06 - Linear Algebra: Vectors, Matrices & Eigenvalues

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