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MATH 110: Abstract Linear Algebra

Table of Contents

Notes for Ken Ribet’s Fall 2025 course. A work in progress.

Lecture 2 (August 29, 2025)

  • What is a field? Describe the field of complex numbers.
  • What axioms define a vector space?
    • Explicitly describe the distributive laws, with respect to maps.
  • What is a function space?

Lecture 3 (September 3, 2025)

  • How can polynomials be represented as a vector space of $n$-tuples?
    • What is $\bigcup_n \mathbb F^n$?
    • What is the degree of the polynomial 0?
  • What is the null space of a matrix?

Lecture 4 (September 5, 2025)

  • In general, what is a subspace of a vector space?
  • What is the smallest subspace containing a list of vectors? Prove it.
  • What is the sum of subspaces?

    Conditions for Subspace

    $U \subseteq V$ is a subspace of $V$ if $U$ is closed under the same addition and multiplication operations, and $U$ contains the additive identity of $V$.

Lecture 5 (September 8, 2025)

  • What is a direct sum of subspaces?
  • What is the smallest subspace containing both $U$ and $V$, where $U \subseteq W$ and $V \subseteq W$?
  • When is a linear map injective?
  • When are $X$ and $Y$ complementary subspaces of $V$?

    Direct Sum of Subspaces

    Definition (Direct Sum). If $U \oplus V$ is a direct sum, then every vector in $U + V$ can be uniquely represented as $u + v$, where $u \in U$ and $v \in V$.

    Conditions for Direct Sum of Subspaces

    Condition. $U+V$ is a direct sum $\iff$ $0 = 0 + 0$ is the only way to represent $0$ as $u+v$, where $u \in U$ and $v \in V$. Injectivity of the summation map.

Condition. $U+V$ is a direct sum $\iff$ $U \cap V = \lbrace 0\rbrace$.

Lecture 6 (September 10, 2025)

  • When does a subspace $X$ have a complement in $V$?
  • What is a linear map from $\mathbb F^l \to v$?
  • What is the span of a list of vector?
  • When is a list of vectors linearly independent?
  • When does a list of vectors span a vector space?
  • What is the basis of a vector space?

    Linear Map

    A linear map respects addition and scalar multiplication:

  • $T(w_1 + w_2) = Tw_1 + Tw_2$
  • $T(\lambda W) = \lambda TW$

    Span of a List of Vectors

    A list of vectors, $v_1, v_2, \ldots, v_l$, where $v_i \in V$, defines a linear map $\mathbb F^l \to V$. The span of this list of vectors is the image of this map.

Lecture 7 (September 12, 2025)

  • What does it mean for a list of vectors to be linearly dependent?
  • How does a linearly independent list relate to the span list of a vector space, $V$?
  • What can be said about subspaces of finite-dimensional vector spaces?
  • How does a spanning list relate to a basis of a vector space?
  • What does it mean for a list of polynomials to be linearly independent?

    Linear Independence

    Definition. $v_1, \ldots, v_n$ are linearly independent if $0 = a_1 v_1 + \ldots + a_n v_n$ necessarily implies that $0 = a_1 = \ldots = a_n$.

Lecture 8 (September 15, 2025)

  • When does a space have a basis?
  • Prove that every subspace of a finite-dimensional space has a complement.
    • How do their dimensions relate? How do their bases relate?

Lecture 9 (September 17, 2025)

  • What is the dimension of a sum of subspaces? Prove it.
  • For a linear map, $T$, when is there a nullity? What is a nullity? What is the nullity of $T$ when $T$ is injective?
  • What is a rank of a linear map?
  • What is the Rank-Nullity Theorem? Prove it.
  • What is the dimension of a Cartesian product of subspaces? Prove it.

    Properties of Linear Maps

    Proposition. If $T$ is injective and $v_1, \ldots, v_l$ is LI, then $Tv_1, \ldots, Tv_l$ is LI.

    Fundamental Theorem of Linear Algebra

    Theorem (FTLA). $T$ is a linear map $T: V \to W$. If $V$ is finite-dimensional, $\text{range } T$ and $\text{null } T$ are finite-dimensional. $\dim V = \text{nullity of }T + \text{rank of }T$.

Lecture 10 (September 19, 2025)

  • What is an isomorphism? When is a linear map an isomorphism?
  • What is the set-theoretic inverse of a linear map? Is it linear?
  • Consider $T: V \to W$. Is $T$ surjective when $\text{dim} V < \text{dim} W$? When is $T$ injective?
  • How are addition and multiplication defined in $\mathcal L(V, W)$?

Lecture 11 (September 22, 2025)

  • How does the basis of $V$ yield an isomorphism between $\mathcal L(V, W)$ and $W^n$? What is $n$?
  • What is $\text{dim} \mathcal L (V, W)$?
  • What is a ‘choice-free’ isomorphism?
  • How does $T$ correspond to $A$?
  • Explain $\mathcal M(ST) = \mathcal M(S) \mathcal M(T)$.
  • Prove that the $ij$th entry of $\mathcal M(ST)$ is the coefficient of $w_i$, where $T: U \to V$, $S : V \to W$.

Lecture 12 (September 24, 2025)

  • What is the column rank of a matrix? What is the row rank of a matrix?
  • What does full-rank mean?
  • How does the column rank of a matrix relate to the column rank of the transposed matrix? Prove it.
  • If $A$ has a column rank of $c \geq 1$, how can it be written as $A = CR$? Can this construction be generalized?

Lecture 13 (September 26, 2025)

  • How is $\mathcal M(T)$ different in one basis of $V$ versus in another basis of $V$?
  • In general, what is the change of basis map?
  • What are quotient spaces?

Lecture 14 (October 1, 2025)

  • What is the surjective linear map associated with a quotient space, $V/U$?

Rank-Nullity Theorem

If $\pi : V \to V/U$ is surjective, by the Rank-Nullity Theorem, $\dim V = \dim U + \dim V/U$, implying that $\dim V/U = \dim V - \dim U$.

  • What is $v + U$, where $U$ is a subspace of $V$?
  • What does $v - v’ \in U$ imply about $v + U$ and $v’ + U$?
  • What is in equivalence class?
  • What is addition for quotient spaces? What is scalar multiplication for quotient spaces?

Dimensions

If $v_1 + U, \ldots, v_t + U$ is a basis of $V/U$, and $u_1, \ldots, u_d$ is a basis of $U$, then $u_1, \ldots, u_d, v_1, \ldots, v_t$ is a basis of $V$.

Complements

$V = U \oplus X$, $\pi: X \to V/U$, the restriction of $\pi$ to $X$. It’s an isomorphism!

  • Prove that the restriction of $\pi$ to $X$ is an isomorphism.

Linear Maps

  • What is $f: \mathcal L(V/U, W) \to \mathcal L(V, W)$?

Lecture 15 (October 3, 2025)

  • What is $T$ if $T: V \to W$ is identically 0 on $U$?

If $S$ is the unique linear map $V/U \to W$ such that $T = S \circ \pi$, then the range of $S$ is the range of $T$.

  • What is the null space of $S$?

  • Prove that $\tilde T$ is injective.

  • What map exists on $\mathcal L(X, W)$ if $\pi: V \to X$ exists?

If $\alpha : W \to Y$, then there exists $\alpha_* : \mathcal L(V, W) \to \mathcal L(V, Y)$. $\mathcal L(V, \cdot)$ is covariant in the second variable.

Dual Spaces

  • What is the vector dual space to $V$? What is its basis?
  • What is the dimension of the dual space?

In general, $\varphi_k(v_j) = \delta_{kj}$, where $\delta$ is the Kronecker-delta.

  • What does $\varphi_k(v)$ calculate? What does $\varphi((x_i, \ldots, x_n)) = x_k$ calculate?