[Axler] The Fundamental Theorem of Linear Maps

Essential Linear Algebra Concepts Summary

Based on Linear Algebra Done Right by Sheldon Axler

This post summarizes the key mathematical concepts from Sheldon Axler's lecture series on Linear Maps, Null Spaces, Ranges, and Matrices.

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1. Vector Spaces & Linear Maps

Core Definition

• Vector Space ($V, W$): A set of vectors over a scalar field $F$ (where $F$ is either $\mathbb{R}$ for real numbers or $\mathbb{C}$ for complex numbers).
• Linear Map ($T$): A function $T : V \to W$ that satisfies two key properties:
  • Additivity: $T(u + v) = T(u) + T(v)$ for all $u, v \in V$.
  • Homogeneity: $T(\lambda u) = \lambda T(u)$ for all $\lambda \in F$ and $u \in V$.

Examples of Linear Maps

• Zero Map ($0$): Maps every vector to the zero vector ($0v = 0$).
• Identity Map ($I$): Maps every vector to itself ($Iv = v$).
• Differentiation ($D$): Maps a polynomial $p$ to its derivative $p'$ ($D(p) = p'$).
• Integration: Maps a polynomial to its integral value (e.g., $\int_{0}^{1} p(x) \, dx$).
• Backward Shift: Applied to sequences, shifts terms to the left (e.g., $(x_1, x_2, \dots) \mapsto (x_2, x_3, \dots)$).

Operations on Linear Maps

• $\mathcal{L}(V, W)$: The set of all linear maps from $V$ to $W$. This set itself forms a vector space.
• Product (Composition): If $T : U \to V$ and $S : V \to W$, the product $ST$ is defined by $(ST)(u) = S(T(u))$.
  • Associativity: $(ST)R = S(TR)$.
  • Distributivity: $S(T_1 + T_2) = ST_1 + ST_2$.
  • Non-Commutativity: Generally, $ST \ne TS$.

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2. Null Spaces and Ranges

Null Space (Kernel)

The Null Space of $T$ (denoted $\operatorname{null} T$) is the set of all vectors in $V$ that map to zero.

$$ \operatorname{null} T = \{ v \in V : T(v) = 0 \} $$

• Property: The null space is a subspace of $V$.
• Injectivity: $T$ is injective (one-to-one) if and only if $\operatorname{null} T = \{0\}$.

Range (Image)

The Range of $T$ (denoted $\operatorname{range} T$) is the set of all possible outputs in $W$.

$$ \operatorname{range} T = \{ T(v) : v \in V \} $$

• Property: The range is a subspace of $W$.
• Surjectivity: $T$ is surjective (onto) if $\operatorname{range} T = W$.

The Fundamental Theorem of Linear Maps

(Rank-Nullity Theorem)

If $V$ is finite-dimensional, then:

$$ \dim V = \dim(\operatorname{null} T) + \dim(\operatorname{range} T) $$

Key Consequences

• Map to Smaller Dimension: If $\dim V > \dim W$, then $T$ cannot be injective.
  (A homogeneous system with more variables than equations has non-zero solutions).
• Map to Larger Dimension: If $\dim V < \dim W$, then $T$ cannot be surjective.
  (An inhomogeneous system with more equations than variables has no solution for some constant terms).

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3. The Matrix of a Linear Map

Matrix Basics

• Matrix ($A$): A rectangular array of scalars with $m$ rows and $n$ columns.
• Notation: $A_{j,k}$ denotes the entry in row $j$ and column $k$.

Constructing the Matrix

To find the matrix of a linear map $T : V \to W$ (denoted $\mathcal{M}(T)$) with respect to bases $v_1, \dots, v_n$ of $V$ and $w_1, \dots, w_m$ of $W$:

1. Compute $T(v_k)$ for each basis vector of $V$.
2. Write $T(v_k)$ as a linear combination of the basis vectors of $W$: $$ T(v_k) = A_{1,k}w_1 + \dots + A_{m,k}w_m $$
3. The coefficients $A_{1,k}, \dots, A_{m,k}$ form the $k$-th column of the matrix.

Algebra of Matrices

• Matrix Addition: Defined by adding corresponding entries. Corresponds to the sum of linear maps: $$ \mathcal{M}(S + T) = \mathcal{M}(S) + \mathcal{M}(T) $$
• Scalar Multiplication: Defined by multiplying each entry by a scalar. Corresponds to scalar multiplication of maps: $$ \mathcal{M}(\lambda T) = \lambda \mathcal{M}(T) $$
• Matrix Space ($F^{m,n}$): The set of all $m \times n$ matrices forms a vector space with dimension $mn$.

The Fundamental Theorem of Linear Maps

Based on Linear Algebra Done Right by Sheldon Axler (Chapter 3)

This theorem acts as the bridge between the domain's dimension and the dimensions of the kernel (null space) and image (range). It is often called the Rank-Nullity Theorem.

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1. The Statement

Theorem

Suppose $V$ is a finite-dimensional vector space and $T \in \mathcal{L}(V, W)$. Then $\range T$ is finite-dimensional and:

$$ \dim V = \dim(\null T) + \dim(\range T) $$

Intuition: When a linear map $T$ acts on $V$, it splits $V$ into two parts:
1. The part that gets sent to $\mathbf{0}$ (the Null Space).
2. The part that forms the image in $W$ (the Range).
The theorem states that the sum of the dimensions of these two parts equals the dimension of the original space.

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2. The Proof (Axler's Approach)

The strategy is to start with a basis for $\null T$, extend it to a basis for $V$, and then use the extension to construct a basis for $\range T$.

Step 1: Basis for Null Space
Let $u_1, \dots, u_m$ be a basis of $\null T$. Thus, $\dim(\null T) = m$.

Step 2: Extend to Basis of V
Since $u_1, \dots, u_m$ is a linearly independent list in $V$, we can extend it to a basis of $V$ (by the Basis Extension Theorem).
Let $u_1, \dots, u_m, v_1, \dots, v_n$ be a basis of $V$. Thus, $\dim V = m + n$.

Step 3: Construct Basis for Range
We claim that $Tv_1, \dots, Tv_n$ is a basis of $\range T$.

(a) Show they span $\range T$:
For any $v \in V$, we can write $v$ as a linear combination of the basis vectors:
$v = a_1 u_1 + \dots + a_m u_m + b_1 v_1 + \dots + b_n v_n$.
Applying $T$:
$Tv = T(\sum a_i u_i) + T(\sum b_j v_j)$.
Since $u_i \in \null T$, $T(u_i) = 0$. Thus, $Tv = b_1 Tv_1 + \dots + b_n Tv_n$.
Therefore, $Tv_1, \dots, Tv_n$ spans $\range T$.

(b) Show they are Linearly Independent:
Suppose $c_1 Tv_1 + \dots + c_n Tv_n = 0$.
By linearity, $T(c_1 v_1 + \dots + c_n v_n) = 0$.
This implies $c_1 v_1 + \dots + c_n v_n \in \null T$.
Therefore, this vector can be written as a combination of $u$'s:
$c_1 v_1 + \dots + c_n v_n = d_1 u_1 + \dots + d_m u_m$.
Moving terms to one side:
$c_1 v_1 + \dots + c_n v_n - d_1 u_1 - \dots - d_m u_m = 0$.
Since the full list $(u_1, \dots, u_m, v_1, \dots, v_n)$ is a basis of $V$, it is linearly independent. Thus, all coefficients, including all $c_j$, must be zero.

Conclusion:
$Tv_1, \dots, Tv_n$ is a basis of $\range T$, so $\dim(\range T) = n$.
Finally:
$$ \dim V = m + n = \dim(\null T) + \dim(\range T) $$ Q.E.D.

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3. Key Consequences

This theorem provides a powerful tool to determine injectivity and surjectivity based solely on dimension.

1. Map to a Smaller Dimension

If $\dim V > \dim W$, then no linear map from $V$ to $W$ is injective.

• Reason: $\dim(\null T) = \dim V - \dim(\range T) \ge \dim V - \dim W > 0$.
  Since $\dim(\null T) > 0$, the null space contains nonzero vectors.
• Example: A homogeneous system with more variables than equations must have non-zero solutions.

2. Map to a Larger Dimension

If $\dim V < \dim W$, then no linear map from $V$ to $W$ is surjective.

• Reason: $\dim(\range T) = \dim V - \dim(\null T) \le \dim V < \dim W$.
  The range cannot cover the entire space $W$.