1[Recipe] Prove $T$ is linear
Take arbitrary $p, q$ and a scalar $\lambda$. Show $T(p + q) = T(p) + T(q)$ AND $T(\lambda p) = \lambda T(p)$. Start each line from the left side.
2[Trap] "$E(p_1) + E(p_2) \in \mathbb{R}^2$, so $E$ is linear."
Not linearity. That only says the outputs land in $\mathbb{R}^2$. Linearity compares $E(p_1 + p_2)$ with $E(p_1) + E(p_2)$, and $E(\lambda p)$ with $\lambda E(p)$.
3[Recipe] Get your hands on a general $p \in \mathcal{P}_3(\mathbb{R})$
Write $p = a_0 + a_1x + a_2x^2 + a_3x^3$. Every condition on $p$ (like $p(1) = 0$) becomes a linear equation in $a_0, \dots, a_3$.
4[Recipe] Basis of $\operatorname{null} T$ when $T$ is defined on $\mathcal{P}_n$
General $p$, impose $Tp = 0$, solve for the $a_i$, write $p$ as (free variable) times (fixed polynomial) + ... Those fixed polynomials are the basis. Count-check with rank-nullity first.
5[Recipe] Show $T : V \to \mathbb{F}^n$ is surjective
For an arbitrary target $(a_1, \dots, a_n)$, name an input that maps to it. Or show $\dim \operatorname{range} T = n$.
6[Trap] Answering "find a basis for $U$" with $U = \{p : \dots\}$
That's a description, not a basis. A basis is a LIST $u_1, \dots, u_k$ of vectors in $U$ that is LI and spans $U$.
7[Prop] Rank-nullity (Prop 24)
$\dim V = \dim \operatorname{null} T + \dim \operatorname{range} T$ ($V$ finite-dim).
8[Prop] Prop 23: injective test
$T$ is injective iff $\operatorname{null} T = \{0\}$.
9[Def] Affine subset (Def 22)
$A = v + U = \{v + u : u \in U\}$ for some $v \in V$ and subspace $U$. A subspace shifted off the origin.
10[Def] Quotient space $V/U$ (Def 24)
The set of all affine subsets $v + U$ parallel to $U$, with $(v + U) + (w + U) = (v + w) + U$ and $\lambda(v + U) = \lambda v + U$.
11[Def] The zero vector of $V/U$
$0 + U = U$. So $v + U$ is zero in $V/U$ exactly when $v \in U$.
12[Prop] Prop 39: the three equivalent conditions
(i) $v - w \in U$. (ii) $v + U = w + U$. (iii) $(v + U) \cap (w + U) \neq \emptyset$. Two cosets are equal or disjoint.
13[Recipe] Check whether $v + U = w + U$
Check $v - w \in U$ (Prop 39).
14[Why] Why do the operations on $V/U$ need a well-defined check?
A coset has many names: $v_1 + U = v_2 + U$ with $v_1 \neq v_2$. The answer can't depend on which name you picked.
15[Recipe] Prove a formula on $V/U$ is well defined
Start from the hypothesis $v + U = w + U$. Prop 39 gives $v - w \in U$. End at: the formula gives the same output for $v$ and $w$.
16[Def] Quotient map $\pi$ (Def 25)
$\pi : V \to V/U$, $\pi(v) = v + U$. Linear, surjective, and $\operatorname{null} \pi = U$.
17[Prop] Prop 40: $\dim V/U$
$\dim V/U = \dim V - \dim U$ (finite-dim). Proof: rank-nullity on $\pi$.
18[Prop] Prop 41 (First Isomorphism Theorem)
$\overline{T} : V/\operatorname{null} T \to \operatorname{range} T$, $\overline{T}(v + \operatorname{null} T) = Tv$, is an isomorphism, and $T = \iota \circ \overline{T} \circ \pi$.
19[Recipe] Prop 41's proof: the four checks, in order
1 well-defined (via Prop 39). 2 linear. 3 injective (output 0 means $v \in \operatorname{null} T$, so $v + \operatorname{null} T$ is the zero coset). 4 surjective (every $Tv$ is hit by $v + \operatorname{null} T$).
20[Recipe] Find a basis of $V/U$
Take a basis $u_1, \dots, u_m$ of $U$. Extend it to a basis of $V$ with $w_1, \dots, w_k$. Then $w_1 + U, \dots, w_k + U$ is a basis of $V/U$.
21[Prop] Prop 42 (induced map)
If $T(U_1) \subseteq U_2$, then $\overline{T}(v + U_1) = Tv + U_2$ is a well-defined linear map $V/U_1 \to W/U_2$.
22[Why] Which hypothesis makes Prop 42 well defined, and how?
$T(U_1) \subseteq U_2$. If $v - w \in U_1$, then $Tv - Tw = T(v - w) \in U_2$, so $Tv + U_2 = Tw + U_2$.
23[Prop] Prop 43 (Second Isomorphism Theorem)
$U/(U \cap W) \cong (U + W)/W$, via $u + (U \cap W) \mapsto u + W$.
24[Prop] Prop 44 (Third Isomorphism Theorem)
If $W \subseteq U \subseteq V$, then $(V/W)/(U/W) \cong V/U$.
25[Ex] Example 43: differentiation and a quotient
$D : \mathcal{P}(\mathbb{R}) \to \mathcal{P}(\mathbb{R})$ is surjective with null space the constants $\mathbb{R}$. So $\mathcal{P}(\mathbb{R})/\mathbb{R} \cong \mathcal{P}(\mathbb{R})$ via $p + \mathbb{R} \mapsto p'$. The inverse is the indefinite integral.
26[Ex] In $\mathbb{R}^2/U$ with $U = \operatorname{Span}((0,1))$: is $(1,2) + U = (1,4) + U$?
Yes. $(1,2) - (1,4) = (0,-2) \in U$. The quotient can't see the $y$-coordinate.
27[T/F] In $V/U$, $v + U = 0$ only when $v = 0$.
False. $v + U = 0$ whenever $v \in U$.
28[T/F] If $\dim V = 5$ and $\dim U = 2$, then $\dim V/U = 3$.
True (Prop 40).
29[T/F] The quotient map $\pi : V \to V/U$ is injective.
False unless $U = \{0\}$. Its null space is $U$.
30[Def] Dual space $V'$ (Def 26)
$V' = \mathcal{L}(V, \mathbb{F})$: all linear maps $\varphi : V \to \mathbb{F}$ (linear functionals).
31[Def] A linear functional, intuitively
A linear measurement of a vector. $\varphi(v)$ is one number about $v$.
32[Prop] $\dim V'$
$\dim V' = \dim V$ when $V$ is finite-dim.
33[Def] Dual basis (Def 27)
For a basis $v_1, \dots, v_n$ of $V$: $\varphi_j(v_k) = \delta_{jk}$, which is 1 if $j = k$ and 0 otherwise.
34[Recipe] What does $\varphi_j$ do to $v = a_1v_1 + \dots + a_nv_n$?
Reads off the $j$-th coordinate: $\varphi_j(v) = a_j$.
35[Prop] Prop 45
The dual basis $\varphi_1, \dots, \varphi_n$ is a basis of $V'$.
36[Why] Prop 45: why is it enough to prove LI?
$\dim V' = \dim V = n$ and the list has length $n$. Right length + LI means basis.
37[Recipe] Prove $\varphi_1, \dots, \varphi_n$ is LI
Suppose $\lambda_1\varphi_1 + \dots + \lambda_n\varphi_n = 0$ (the zero functional). Plug in $v_k$: the left side gives $\lambda_k$, so every $\lambda_k = 0$.
38[Ex] Example 44: the dual of the standard basis of $\mathbb{F}^n$
The row vectors $(1, 0, \dots, 0), \dots, (0, \dots, 0, 1)$. Functionals on $\mathbb{F}^n$ are row vectors.
39[T/F] The dual basis vector $\varphi_1$ depends only on $v_1$.
False. It depends on the whole basis. For $(1,0), (0,1)$: $\varphi_1(x,y) = x$. For $(1,0), (1,1)$: $\varphi_1(x,y) = x - y$.
40[Def] Dual map $T'$ (Def 28): domain, codomain, formula
For $T : V \to W$: $T' : W' \to V'$, $T'(\varphi) = \varphi \circ T$. Do $T$, then measure.
41[Recipe] Compute $T'\varphi$
$(T'\varphi)(v) = \varphi(T(v))$. Plug in a general $v$ and simplify. The answer is a functional on $V$ (a formula or row vector).
42[Trap] Treating $T'$ like $T^{-1}$
$T^{-1}$ eats vectors in $W$. $T'$ eats functionals on $W$ and returns functionals on $V$. $T'$ exists for every linear $T$, invertible or not.
43[Trap] Writing $T'(T(v))$ or $(ST)'v$
Type error. $T'$ only takes functionals. Write $(T'\varphi)(v)$.
44[T/F] For $T : V \to W$, the dual map goes $V' \to W'$.
False. $T' : W' \to V'$. The direction flips.
45[T/F] $T'$ exists only when $T$ is invertible.
False. Every linear map has a dual map.
46[Ex] Example 45: $T$ = multiply by $x$ on $\mathcal{P}(\mathbb{R})$, $\varphi(p) = p(2)$. Find $T'\varphi$.
$T'(\varphi)(p) = \varphi(xp) = 2p(2)$, so $T'\varphi = 2\varphi$.
47[Ex] $T : \mathbb{R}^2 \to \mathbb{R}$, $T(x, y) = x + 3y$, and $\ell(t) = 2t$. Find $T'\ell$.
$(T'\ell)(x, y) = \ell(x + 3y) = 2x + 6y$. Row vector $(2, 6)$.
48[Recipe] Show two functionals in $V'$ are equal
Show they give the same number at every $v \in V$ (or at every vector of a basis).
49[Prop] Prop 47
$T \mapsto T'$ is linear: $(\lambda T_1 + T_2)' = \lambda T_1' + T_2'$. And it reverses products: $(ST)' = T'S'$.
50[Why] Prop 47 (ii): why does $(ST)' = T'S'$ flip the order?
$(ST)'\varphi = \varphi \circ S \circ T = (S'\varphi) \circ T = T'(S'\varphi)$. $S'$ acts first.
51[Prop] Prop 49
In the dual bases, $\mathcal{M}(T')$ is the transpose of $\mathcal{M}(T)$.
52[Ex] $T : \mathbb{R}^3 \to \mathbb{R}^2$. Sizes of $\mathcal{M}(T)$ and $\mathcal{M}(T')$?
$\mathcal{M}(T)$ is $2 \times 3$. $\mathcal{M}(T')$ is $3 \times 2$.
53[Ex] Example 46: dual of $A$ with rows $(1, 2, 3)$ and $(4, 5, 6)$
$A' : (\mathbb{R}^2)' \to (\mathbb{R}^3)'$, $(x\ \ y) \mapsto (x\ \ y)A$. Its matrix is $A^T$, rows $(1, 4), (2, 5), (3, 6)$.
54[Prop] Props 47 + 49 give which matrix rule?
$(AB)^T = B^TA^T$.
55[Def] Annihilator $U^0$ (Def 29)
$U^0 = \{\varphi \in V' : \varphi(u) = 0 \text{ for all } u \in U\}$. The measurements that read 0 on all of $U$.
56[Prop] $\dim U^0$
$\dim U + \dim U^0 = \dim V$ (finite-dim).
57[Recipe] Find $U^0$ in $\mathbb{R}^n$
Look for row vectors $c$ with $c \cdot u_i = 0$ for every vector $u_i$ in a basis of $U$. Solve the system.
58[T/F] $\{0\}^0 = V'$ and $V^0 = \{0\}$.
True.
59[Prop] Prop 50
$\operatorname{null} T' = (\operatorname{range} T)^0$ and $\operatorname{range} T' = (\operatorname{null} T)^0$.
60[Prop] Prop 51
$T$ injective iff $T'$ surjective. $T$ surjective iff $T'$ injective.
61[Recipe] Asked to PROVE "$T$ surjective implies $T'$ injective" (no quoting Prop 51)
Prop 23 on $T'$: suppose $T'\varphi = 0$. Unfold Def 28 at every $v$. Use surjectivity to reach every $w \in W$.
62[T/F] On an exam you can quote Prop 51 even when the question asks you to prove it.
False. You may quote results unless the problem asks you to prove them.
63[Prop] Prop 46 (double dual)
$j_V(v)(\varphi) = \varphi(v)$ defines an injective linear map $V \to V''$. It's an isomorphism when $V$ is finite-dim.
64[Why] What Prop 46 says in words
If $\varphi(v) = \varphi(w)$ for every $\varphi \in V'$, then $v = w$. A vector is determined by all its linear measurements.
65[Prop] Prop 48
$T'' \circ j_V = j_W \circ T$ for every linear $T : V \to W$.
66[T/F] Every linear functional $\varphi : V \to \mathbb{F}$ is either 0 or surjective.
True. Its range is a subspace of $\mathbb{F}$, so it's $\{0\}$ or $\mathbb{F}$.
67[Def] Invariant subspace (Def 30)
$U$ is invariant under $T$ if $T(U) \subseteq U$.
68[Recipe] Check that $U$ is invariant
Check $Tu_i \in U$ for each vector $u_i$ of a basis of $U$.
69[Ex] Example 48: subspaces that are always invariant
$\{0\}$, $V$, $\operatorname{null} T$, $\operatorname{range} T$.
70[Def] Eigenvalue and eigenvector (Def 31)
$\lambda$ is an eigenvalue if $Tv = \lambda v$ for some $v \neq 0$. That $v$ is an eigenvector.
71[Def] Eigenspace (Def 32)
$E(\lambda, T) = \operatorname{null}(T - \lambda I)$.
72[Recipe] Is $\lambda$ an eigenvalue?
Is $\operatorname{null}(T - \lambda I) \neq \{0\}$? Same as: $T - \lambda I$ is not injective.
73[Trap] "The only solution is $v = 0$, so $\lambda$ is an eigenvalue."
Backwards. $v = 0$ always solves $Tv = \lambda v$. Only-zero means $\lambda$ is NOT an eigenvalue.
74[Prop] Prop 53
Every operator on a nonzero finite-dim COMPLEX vector space has an eigenvector.
75[Prop] Prop 54
Eigenvectors for distinct eigenvalues are linearly independent.
76[Prop] Prop 55
An operator on $V$ has at most $\dim V$ distinct eigenvalues.
77[Ex] Eigenvalues of $R(x, y) = (-y, x)$
None over $\mathbb{R}$. Over $\mathbb{C}$: $i$ and $-i$.
78[Prop] $2 \times 2$ characteristic polynomial of $A$ with rows $(a, b), (c, d)$
$t^2 - (a + d)t + (ad - bc)$. Its roots are the eigenvalues.
79[Why] Why is $\operatorname{range}(T - \lambda I)$ invariant under $T$?
$T$ commutes with $T - \lambda I$.
80[T/F] A nonzero $v$ is an eigenvector of $T$ iff $\operatorname{Span}(v)$ is invariant.
True.
81[Def] Upper-triangular (Def 33)
$a_{ij} = 0$ for $i > j$: every entry below the diagonal is 0.
82[Prop] Prop 59
$\mathcal{M}(T)$ is upper-triangular iff $Tv_j \in \operatorname{Span}(v_1, \dots, v_j)$ for each $j$ iff each $\operatorname{Span}(v_1, \dots, v_j)$ is invariant.
83[Prop] Prop 60
If $\mathcal{M}(T)$ is upper-triangular, the eigenvalues of $T$ are exactly its diagonal entries.
84[Prop] Prop 61
Over $\mathbb{C}$ ($V \neq 0$, finite-dim), some basis makes $\mathcal{M}(T)$ upper-triangular.
85[Recipe] Prop 61's proof: the moves
Induction on $\dim V$. Base: $\dim V = 1$. Step: eigenvalue $\lambda$ (Prop 53), $U = \operatorname{range}(T - \lambda I)$, invariant with $\dim U < \dim V$, IH on $T|_U$, extend to a basis of $V$, $Tv_j = (T - \lambda I)v_j + \lambda v_j$, finish with Prop 59.
86[Why] In Prop 61, why is $\dim \operatorname{range}(T - \lambda I) < \dim V$?
$\lambda$ is an eigenvalue, so $\operatorname{null}(T - \lambda I) \neq \{0\}$. Then rank-nullity.
87[Trap] Skipping a wordy multi-part proof
The parts ARE the proof skeleton. Write one line per part before deciding to skip. Part (a) is usually cheap points.
88[Recipe] Induction proof layout
Base case (smallest $n$). Inductive hypothesis (assume for all smaller dims / for $n$). Step (show for the next one, and point at the line that uses the IH).
89[Def] Diagonalizable (Def 34)
Some basis makes $\mathcal{M}(T)$ diagonal. Same as: $V$ has a basis of eigenvectors.
90[Prop] Prop 62: four equivalent conditions
Diagonalizable. Eigenbasis exists. $V = E(\lambda_1, T) \oplus \dots \oplus E(\lambda_m, T)$. $\dim V = \sum \dim E(\lambda_i, T)$.
91[Prop] Prop 63
$\dim V$ distinct eigenvalues implies diagonalizable. Not the converse.
92[Prop] Prop 64
In an eigenbasis, $T^k(a_1v_1 + \dots + a_nv_n) = a_1\lambda_1^kv_1 + \dots + a_n\lambda_n^kv_n$.
93[Def] Geometric multiplicity of $\lambda$
$\dim E(\lambda, T)$.
94[T/F] Diagonalizable implies $\dim V$ distinct eigenvalues.
False. $I$ is diagonal with one eigenvalue.
95[T/F] The matrix with rows $(1, 1), (0, 1)$ is diagonalizable.
False. Only eigenvalue 1 and $\dim E(1) = 1 < 2$.
96[T/F] Diagonalizable over $\mathbb{C}$ implies diagonalizable over $\mathbb{R}$.
False. Rotation $R(x, y) = (-y, x)$.
97[Def] $p(T)$ (Def 35)
If $p(x) = a_nx^n + \dots + a_1x + a_0$, then $p(T) = a_nT^n + \dots + a_1T + a_0I$.
98[Def] Minimal polynomial $m_T$ (Def 36)
The monic polynomial of smallest degree with $m_T(T) = 0$.
99[Recipe] Compute $m_T$
Find the first $k$ with $T^k \in \operatorname{Span}(I, T, \dots, T^{k-1})$. Solve $T^k = c_0I + \dots + c_{k-1}T^{k-1}$ by matching entries. Then $m_T = x^k - c_{k-1}x^{k-1} - \dots - c_0$.
100[Trap] The degree-1 test for $m_T$
Ask "is $T = cI$?", not "is $T = cT$?". The degree-2 test is $T^2 = aT + bI$.
101[Prop] Prop 65
$q(T) = 0$ iff $m_T$ divides $q$.
102[Prop] Prop 66
The eigenvalues of $T$ are exactly the zeros of $m_T$ in $\mathbb{F}$.
103[Prop] Prop 67
$\deg m_T \le \dim V$.
104[Prop] Prop 68
$T$ is diagonalizable iff $m_T = (x - \lambda_1) \cdots (x - \lambda_m)$ with distinct $\lambda_i \in \mathbb{F}$.
105[Prop] Invertibility from $m_T$ (Axler 5.32)
$T$ is invertible iff $m_T(0) \neq 0$ (the constant term isn't 0).
106[Recipe] Write $T^{-1}$ as a polynomial in $T$
Start from $m_T(T) = 0$ with constant term $a_0 \neq 0$. Move $a_0I$ to one side, factor $T$ out of the rest, divide by $-a_0$.
107[Trap] "Eigenvalues exist, so $T$ is invertible."
No. Over $\mathbb{R}$ there may be none, and 0 can be one. Invertible iff 0 is NOT an eigenvalue.
108[Trap] Writing $m_T^{-1}$ for the inverse
$m_T$ is a polynomial, not a map. The inverse is $T^{-1}$, written as a polynomial in $T$.
109[Ex] $m_I$ on any $V \neq 0$
$x - 1$ (Example 58).
110[Ex] $m_N$ for $N$ with rows $(0, 1), (0, 0)$
$x^2$.
111[Ex] $m_R$ for $R(x, y) = (-y, x)$ on $\mathbb{R}^2$
$x^2 + 1$, since $R^2 = -I$ and $R$ is not a multiple of $I$.
112[T/F] The minimal polynomial always equals the characteristic polynomial.
False. $I$ on $\mathbb{C}^2$: $m = x - 1$, characteristic $(x - 1)^2$.
113[T/F] If $T^3 = I$ on a complex space, $T$ is diagonalizable.
True. $m_T$ divides $x^3 - 1$, which has 3 distinct roots in $\mathbb{C}$ (Props 65, 68).
114[T/F] If $m_T(0) = 0$, then 0 is an eigenvalue of $T$.
True (Prop 66).
115[Def] Generalized eigenspace (Def 37)
$G(\lambda, T) = \bigcup_k \operatorname{null}(T - \lambda I)^k$. Its nonzero vectors are generalized eigenvectors.
116[Prop] Prop 69
$\{0\} = \operatorname{null} T^0 \subseteq \operatorname{null} T^1 \subseteq \operatorname{null} T^2 \subseteq \dots$, and once $\operatorname{null} T^m = \operatorname{null} T^{m+1}$, it stays the same forever.
117[Prop] Prop 70
$G(\lambda, T) = \operatorname{null}(T - \lambda I)^{\dim V}$.
118[Prop] Prop 71
$G(\lambda, T) \neq \{0\}$ iff $\lambda$ is an eigenvalue.
119[Prop] Prop 73
Over $\mathbb{C}$: $V = G(\lambda_1, T) \oplus \dots \oplus G(\lambda_m, T)$.
120[Def] Algebraic multiplicity (Def 38)
$\dim G(\lambda, T)$.
121[Prop] Geometric vs algebraic multiplicity
$\dim E(\lambda, T) \le \dim G(\lambda, T)$. Diagonalizable iff they're equal for every eigenvalue.
122[Prop] Prop 74 (Cayley-Hamilton)
Over $\mathbb{C}$, with distinct eigenvalues $\lambda_i$ and algebraic multiplicities $d_i$: $q(x) = (x - \lambda_1)^{d_1} \cdots (x - \lambda_m)^{d_m}$ satisfies $q(T) = 0$.
123[Def] Nilpotent (Def 39)
$N^k = 0$ for some $k$.
124[Prop] Facts about a nilpotent $N$
$N^{\dim V} = 0$, and 0 is its only eigenvalue.
125[T/F] Over $\mathbb{C}$, the algebraic multiplicities add up to $\dim V$.
True (Prop 73).
126[T/F] Every generalized eigenvector is an eigenvector.
False. In $J$ with rows $(0, 1), (0, 0)$, $e_2$ is a generalized eigenvector for 0 but $Je_2 = e_1 \neq 0$.
127[Def] Jordan block (Def 40)
Square block with $\lambda$ on the diagonal, 1s just above the diagonal, 0 elsewhere. The $\lambda$'s of different blocks don't have to be distinct.
128[Prop] Prop 76
Over $\mathbb{C}$, every operator has a basis in which its matrix is in Jordan normal form.
129[Prop] Prop 77 (book's formula)
Number of Jordan blocks for $\lambda$ of size exactly $k$ = $2\dim\operatorname{null}(T - \lambda I)^k - \dim\operatorname{null}(T - \lambda I)^{k-1} - \dim\operatorname{null}(T - \lambda I)^{k+1}$.
130[Recipe] Number of Jordan blocks for $\lambda$ of size at least $k$
$\dim\operatorname{null}(T - \lambda I)^k - \dim\operatorname{null}(T - \lambda I)^{k-1}$.
131[Recipe] Read multiplicities off a Jordan form
Number of blocks for $\lambda$ = geometric multiplicity. Total size of those blocks = algebraic multiplicity. Largest block size = exponent of $(x - \lambda)$ in $m_T$.
132[Ex] Blocks $J_2(3), J_1(3)$: characteristic poly, minimal poly, diagonalizable?
Characteristic $(x - 3)^3$. Minimal $(x - 3)^2$. Geometric 2, algebraic 3, so not diagonalizable.
133[T/F] A Jordan form whose blocks all have size 1 is diagonal.
True.
134[Trap] Where the 1s go in a Jordan form
Only just above the diagonal, inside a block. Never between two blocks.