Mathematics Linear Algebra Syllabus

Linear Algebra syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Linear Algebra in Mathematics, not a summary of it.

1. 1

   Vectors

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

2. 2

   Matrices

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

3. 3

   Determinants

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

4. 4

   Vector Spaces

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

5. 5

   Eigenvalues and Eigenvectors

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

6. 6

   Linear Transformations

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

7. 7

   Inner Product Spaces

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

8. 8

   Orthogonality

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

9. 9

   Diagonalization

   overview

   Examined as a single unit within Linear Algebra — no further topic split in the official outline.

Linear Algebra flashcards for Mathematics

19 of 51 cards from the Linear Algebra deck — real questions with worked answers.

1. What is a vector space over a field $F$?

   A set $V$ with vector addition and scalar multiplication satisfying: associativity and commutativity of addition, an additive identity $\vec{0}$, additive inverses, and compatibility/distributivity of scalar multiplication ($a(\vec{u}+\vec{v})=a\vec{u}+a\vec{v}$, $(a+b)\vec{v}=a\vec{v}+b\vec{v}$, $a(b\vec{v})=(ab)\vec{v}$, $1\vec{v}=\vec{v}$).

2. What does it mean for a set of vectors $\{\vec{v}_1,\dots,\vec{v}_n\}$ to be linearly independent?

   The only solution to $c_1\vec{v}_1+c_2\vec{v}_2+\cdots+c_n\vec{v}_n=\vec{0}$ is $c_1=c_2=\cdots=c_n=0$. No vector is a linear combination of the others.

3. What is a basis of a vector space $V$?

   A set of vectors that is both linearly independent and spans $V$. Every vector in $V$ can be written uniquely as a linear combination of basis vectors.

4. Define the dimension of a vector space.

   The number of vectors in any basis of the space. All bases of a finite-dimensional space have the same number of elements.

5. What is the span of a set of vectors?

   The set of all possible linear combinations of those vectors: $\operatorname{span}\{\vec{v}_1,\dots,\vec{v}_n\}=\{c_1\vec{v}_1+\cdots+c_n\vec{v}_n : c_i\in F\}$.

6. What conditions make $W$ a subspace of vector space $V$?

   $W$ must contain $\vec{0}$, be closed under addition ($\vec{u},\vec{v}\in W \Rightarrow \vec{u}+\vec{v}\in W$), and closed under scalar multiplication ($c\vec{v}\in W$).

7. How is the dot product of $\vec{u}=(u_1,\dots,u_n)$ and $\vec{v}=(v_1,\dots,v_n)$ computed?

   $\vec{u}\cdot\vec{v}=\sum_{i=1}^{n}u_i v_i = u_1 v_1 + u_2 v_2 + \cdots + u_n v_n$.

8. What is the geometric formula relating the dot product to the angle between vectors?

   $\vec{u}\cdot\vec{v}=\|\vec{u}\|\,\|\vec{v}\|\cos\theta$, where $\theta$ is the angle between them.

9. When are two nonzero vectors orthogonal?

   When their dot product is zero: $\vec{u}\cdot\vec{v}=0$, meaning $\theta=90^\circ$.

10. What is the Euclidean norm (length) of a vector $\vec{v}=(v_1,\dots,v_n)$?

    $\|\vec{v}\|=\sqrt{\vec{v}\cdot\vec{v}}=\sqrt{v_1^{2}+v_2^{2}+\cdots+v_n^{2}}$.

11. What is the formula for matrix multiplication entry $(AB)_{ij}$?

    $(AB)_{ij}=\sum_{k=1}^{n}A_{ik}B_{kj}$, the dot product of row $i$ of $A$ with column $j$ of $B$. It is defined only when the columns of $A$ equal the rows of $B$.

12. Is matrix multiplication commutative?

    No. In general $AB \neq BA$. Matrix multiplication is associative and distributive but not commutative.

13. What is the identity matrix $I_n$?

    The $n\times n$ matrix with $1$'s on the main diagonal and $0$'s elsewhere, satisfying $AI=IA=A$. It is the multiplicative identity.

14. Define the transpose $A^{T}$ of a matrix $A$.

    The matrix obtained by swapping rows and columns: $(A^{T})_{ij}=A_{ji}$. Properties: $(A^{T})^{T}=A$ and $(AB)^{T}=B^{T}A^{T}$.

15. What is a symmetric matrix?

    A square matrix equal to its transpose: $A=A^{T}$, i.e. $A_{ij}=A_{ji}$.

16. What is the formula for the inverse of a $2\times 2$ matrix $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

    $A^{-1}=\dfrac{1}{ad-bc}\begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, valid when $\det A = ad-bc \neq 0$.

17. When is a square matrix $A$ invertible (nonsingular)?

    When $\det A \neq 0$, equivalently when its columns are linearly independent, it has full rank, and $A\vec{x}=\vec{0}$ has only the trivial solution.

18. What is the inverse-of-a-product rule?

    $(AB)^{-1}=B^{-1}A^{-1}$ (the order reverses), provided both $A$ and $B$ are invertible.

19. How do you compute the determinant of a $2\times 2$ matrix $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

    $\det = ad - bc$.

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