Linear Algebra - Lecture 13 - Fall, 2026

Overview

The lecture develops linear transformations through their kernels and images, defining nullity and rank and proving the rank-nullity theorem: for a finite-dimensional domain V, nullity(T) + rank(T) = dim(V). Examples in R² and R³ connect these ideas to projections, systems of equations, and matrix multiplication; the lecture also proves that a basis for a subspace can be extended to one for the whole space and relates kernels to injectivity.

Key takeaways

The kernel of a linear transformation is always a subspace of its domain, and its dimension is the transformation’s nullity.
The image is always a subspace of the codomain, and its dimension is the transformation’s rank.
For every linear transformation with a finite-dimensional domain V, rank(T) + nullity(T) = dim(V); a kernel basis extended to a basis of V gives a direct proof.
For T(x, y) = (x, 0), the kernel is the y-axis and the image is the x-axis, so nullity and rank are both 1 in R².
For a linear transformation, being one-to-one is equivalent to having kernel {0}; this follows by applying linearity to T(v₁) - T(v₂).
For a linear operator on a finite-dimensional space, being onto, having full rank, and having zero kernel are equivalent.

Chapters

0:00 Review Problem: Subspaces and Parallel Affine Lines in R²

The subspace defined by 3x - 2y = 0 is a line through the origin with slope 3/2.
The equations 3x - 2y = -2 and 3x - 2y = 2 give parallel affine lines with y-intercepts 1 and -1.
The review also notes the midterm is Wednesday at 11:00 in Gillette Hall, room 205.

4:03 Why a Translate of a Subspace Is an Affine Subspace

For L = v + W, each point in L can be written as v + w for some w in the subspace W.
If x = v + w₁ and y = v + w₂, then (1 - t)x + ty = v + (1 - t)w₁ + tw₂.
Because W is closed under linear combinations, (1 - t)x + ty remains in L for every scalar t.

10:18 The Two Defining Rules for a Linear Transformation

A map T: V → W is linear when T(v₁ + v₂) = T(v₁) + T(v₂) for all v₁, v₂ in V.
Linearity also requires T(cv) = cT(v) for every vector v and scalar c.
These rules apply to maps between potentially different vector spaces V and W.

12:46 Matrix Multiplication Is Linear, but Squaring Is Not

An m-by-n matrix A defines a linear map Tₐ: Rⁿ → Rᵐ by Tₐ(v) = Av.
The example matrix with rows (3, 2), (7, 1), and (-6, 5) maps (x, y) to (3x + 2y, 7x + y, -6x + 5y) in R³.
The function T(x) = x² fails additivity: T(3 + 5) = 64, while T(3) + T(5) = 34.

18:14 The Kernel Is a Subspace of the Domain

The kernel of T is the set of vectors v in V satisfying T(v) = 0.
Linearity implies T(0) = 0, so the zero vector belongs to the kernel.
If v₁ and v₂ are in the kernel, then T(v₁ + v₂) = 0; if v is in the kernel, T(cv) = 0.
The nullity of T is defined as the dimension of its kernel.

23:57 Projection onto the x-Axis: Kernel and Nullity

The map T(x, y) = (x, 0) sends every vector in R² to its projection onto the x-axis.
Its kernel consists of all vectors (0, y), which form the y-axis and are spanned by (0, 1).
Since the kernel is one-dimensional, this projection has nullity 1.

30:10 Kernels from Systems of Equations in R² and R³

For T(x, y) = (x + y, x - y), setting both outputs to zero gives x = y = 0, so the kernel is {0}.
For T(x, y, z) = x + y + z, the kernel satisfies z = -x - y.
That R³ kernel is spanned by (1, 0, -1) and (0, 1, -1), so its nullity is 2.

38:30 The Image of a Linear Transformation Is a Subspace

The image of T consists of output vectors T(v) for v in the domain V.
If w₁ = T(v₁) and w₂ = T(v₂), then w₁ + w₂ = T(v₁ + v₂), so the image is closed under addition.
For any scalar c, cw₁ = T(cv₁); the image also contains zero and is therefore a subspace of W.
The rank of T is the dimension of its image.

42:32 Computing Rank from Projection and Invertible Coordinate Maps

For T(x, y) = (x, 0), the image is the x-axis, so rank(T) = 1; its nullity is also 1.
For T(x, y) = (x + y, x - y), any target (a, b) has preimage ((a + b)/2, (a - b)/2).
That second map has image R² and rank 2, consistent with its zero-dimensional kernel.

47:32 Rank-Nullity Theorem: Extending a Kernel Basis

For a linear transformation from an n-dimensional space V, the theorem states nullity(T) + rank(T) = n.
Let the kernel have dimension k and choose a basis v₁, …, vₖ for it.
Extend that kernel basis with n - k vectors to form a basis of V.

52:12 Rank-Nullity Proof: Images of the Added Basis Vectors

Apply T to the extended basis: the images of the first k vectors vanish because they lie in the kernel.
Every vector in the image is therefore a linear combination of the n - k images of the added basis vectors.
Those n - k images are linearly independent: any zero combination would produce a vector in the kernel and contradict independence of the extended basis.
The images form a basis for the image, giving rank(T) = n - k and nullity(T) = k.

59:27 Extending a Subspace Basis to a Basis of the Whole Space

If w₁, …, wₖ form a basis for a proper subspace W and w lies outside W, adding w preserves linear independence.
A dependence relation with a nonzero coefficient on w would express w as a linear combination of vectors in W, a contradiction.
In an n-dimensional space, keep adding vectors outside the current span until the independent set has n vectors and forms a basis.

1:07:13 Linear Operators and the Finite-Dimensional Isomorphism Criterion

A linear transformation from a vector space V to itself is called a linear operator.
For an operator on an n-dimensional space, rank(T) + nullity(T) = n.
A zero kernel implies rank n and image V; in finite dimensions, the operator is then invertible.

1:09:58 A Linear Map Is One-to-One Exactly When Its Kernel Is Zero

A map is one-to-one when distinct domain vectors have distinct images.
Linearity gives T(v₁) = T(v₂) exactly when v₁ - v₂ lies in the kernel.
If the kernel is {0}, equal outputs force v₁ = v₂; any nonzero kernel vector instead shares the zero output with the zero vector.

1:13:47 Onto Maps, Rank, and Midterm Instructions

A map T: V → W is onto when every vector in W belongs to its image.
For an operator T: V → V, being onto gives rank(T) = dim(V), and rank-nullity then forces a zero kernel.
The midterm permits no electronic devices or calculators; students should bring paper and pencil to Gillette Hall, room 205.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Nathanson math lectures.