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Diffstat (limited to 'mathematics')
| -rw-r--r-- | mathematics/linear-algebra.md | 103 |
1 files changed, 55 insertions, 48 deletions
diff --git a/mathematics/linear-algebra.md b/mathematics/linear-algebra.md index a65f741..4e1162b 100644 --- a/mathematics/linear-algebra.md +++ b/mathematics/linear-algebra.md @@ -75,16 +75,16 @@ Our definition of a vector space leads to some facts: - Proof: by definition, the zero vector exists. - The additive inverse for some $x$ is *unique*. - Proof: exercise -- If $V$ is a vector space over $𝔽$ and $V ≠ \{0\}$, then $V$ is an *infinite set*. (note this only holds over $𝔽$) +- If $V$ is a vector space over $𝔽$ and $V ≠ \\{0\\}$, then $V$ is an *infinite set*. (note this only holds over $𝔽$) - Proof: you can just keep adding things -Example: Let $S = \{(a_1, a_2) | a_1, a_2 ∈ ℝ\}$. +Example: Let $S = \\{(a_1, a_2) | a_1, a_2 ∈ ℝ\\}$. For $(a_1, a_2), (b_1, b_2) ∈ S$ and $c ∈ ℝ$, we define: - $(a_1, a_2) + (b_1, b_2) = (a_1 + b_1, a_2 - b_2)$ - $c(a_1, a_2) = (ca_1, ca_2)$. - This fails commutativity! -Example: Let $S = \{(a_1, a_2) | a_1, a_2 ∈ ℝ\}$. We define: +Example: Let $S = \\{(a_1, a_2) | a_1, a_2 ∈ ℝ\\}$. We define: - $(a_1, a_2) + (b_1, b_2) = (a_1 + b_1)$ - $c(a_1, a_2) = (ca_1, 0)$ - fails zero! @@ -104,7 +104,7 @@ We call $a_1 ... a_n$ the *coefficients* of the linear combination. https://math.stackexchange.com/questions/3492590/linear-combination-span-independence-and-bases-for-infinite-dimensional-vector -Let $S$ be a nonempty subset of a vector space $V$. The **span** of $S$, denoted $span(S)$, is the set consisting of all linear combination of vectors in S. For convenience, we define $span(∅) = \{0\}$. +Let $S$ be a nonempty subset of a vector space $V$. The **span** of $S$, denoted $span(S)$, is the set consisting of all linear combination of vectors in S. For convenience, we define $span(∅) = \\{0\\}$. The span of any subset $S$ of a vector space $V$ is a subspace of $V$. @@ -127,7 +127,7 @@ Let $S$ be a linearly independent subset of a vector space $V$, and let $v ∈ V A basis $B$ for a vector space $V$ is a *linearly independent* subset of $V$ that *spans* $V$. If $B$ is a basis for $V$, we also say that the vectors of $B$ form a basis for $V$. -Let V be a vector space and β = {v_1, ..., v_n} be a subset of V. Then β is a basis for V iff every v ∈ V can be **uniquely expressed** as a linear combination of vectors of β. that is, V can be written in the form v = a_1 u_1 + a_2 u_2 ... a_n u_n for unique scalars a. +Let $V$ be a vector space and $β = {v_1, ..., v_n}$ be a subset of V. Then β is a basis for V iff every $v ∈ V$ can be **uniquely expressed** as a linear combination of vectors of β. that is, V can be written in the form $v = a_1 u_1 + a_2 u_2 ... a_n u_n$ for unique scalars a. If a vector space V is spanned by a finite set S, then some subset of S is a basis of V. So, V has a finite basis. Proof: If $S = ∅$, then $V = span{S} = span{∅} = \span{𝕆}$ in which case ∅ is a basis for $V$. @@ -144,9 +144,9 @@ Theorem 1.4: Let $W$ be a subspace of a finite-dimensional vector space $V$. The ## Linear Transformations -Let $V$ and $W$ be vector spaces (over $F$). +Let $V$ and $W$ be vector spaces (over a field $F$). -Definition: A function $T: V → W$ is a **linear transformation** from $V$ into $W$ if $∀x,y ∈ V, c ∈ F$ we have $T(cx + y) = cT(x) + T(y)$. +A function $T: V → W$ is a **linear transformation** from $V$ into $W$ if $∀x,y ∈ V, c ∈ F$ we have $T(cx + y) = cT(x) + T(y)$. Subsequently: - $T(x + y) = T(x) + T(y)$ - $T(cx) = cT(x)$ @@ -155,9 +155,8 @@ Subsequently: Let $T: V → W$ be a linear transformation. -Definition: -The **kernel** (or null space) $N(T)$ of $T$ is the set of all vectors in $V$ such that $T(x) = 0$: $N(T) = \{ x ∈ V : T(x) = 0 \}$. -The **image** (or range) $R(T)$ of $T$ is the subset of $W$ consisting of all images (under $T$) of elements of $V$: $R(T) = \{ T(x) : x ∈ V \}$ +The **kernel** (or null space) $N(T)$ of $T$ is the set of all vectors in $V$ such that $T(x) = 0$: $N(T) = \\{ x ∈ V : T(x) = 0 \\}$. +The **image** (or range) $R(T)$ of $T$ is the subset of $W$ consisting of all images (under $T$) of elements of $V$: $R(T) = \\{ T(x) : x ∈ V \\}$ Theorem: The kernel $N(T)$ and image $R(T)$ are subspaces of $V$ and $W$, respectively. <details> @@ -170,13 +169,13 @@ Let $x,y ∈ R(T)$ and $c ∈ F$. As $T(0_v) = 0_w$, $0_w ∈ R(T)$. ... </details> -Theorem: If $β = \{ v_1, v_2, ... v_n \}$ is a basis for $V$, then $R(T) = span(\{ T(v_1), T(v_2), ..., T(v_n) \})$. +Theorem: If $β = \\{ v_1, v_2, ... v_n \\}$ is a basis for $V$, then $R(T) = span(\\{ T(v_1), T(v_2), ..., T(v_n) \\})$. <details> <summary>Proof</summary> ... </details> -Definition: If $N(T)$ and $R(T)$ are finite-dimensional, then the **nullity** and **rank** of T are the dimensions of $N(T)$ and $R(T)$, respectively. +If $N(T)$ and $R(T)$ are finite-dimensional, then the **nullity** and **rank** of T are the dimensions of $N(T)$ and $R(T)$, respectively. Rank-Nullity Theorem: If $V$ is *finite-dimensional*, then $dim(V) = nullity(T) + rank(T)$. <details> @@ -184,76 +183,84 @@ Rank-Nullity Theorem: If $V$ is *finite-dimensional*, then $dim(V) = nullity(T) ... </details> -Definition: Recall that a *function* definitionally maps *each* element of its domain to *exactly* one element of its codomain. -A function is **injective** (or one-to-one) iff each element of its domain maps to a *distinct* element of its codomain. -A function **surjective** (or onto) iff each element of the codomain is mapped to by *at least* one element in the domain. +A function is **injective** (or **one-to-one**) iff each element of its domain maps to a *distinct* element of its codomain. +A function is **surjective** (or **onto**) iff each element of the codomain is mapped to by *at least* one element in the domain. A function is **bijective** iff it is surjective and injective. Necessarily, a bijective function is invertible, which will be formally stated & proven later. -Theorem: $T$ is injective iff $N(T) = \{0\}$. +Theorem: $T$ is injective iff $N(T) = \\{0\\}$. <details> <summary>Proof</summary> ... </details> -Theorem: For $V$ and $W$ of equal (finite) dimension: $T$ is injective iff it is surjective. +Theorem: For $V$ and $W$ of equal (and finite) dimension: $T$ is injective iff it is surjective. <details> <summary>Proof</summary> ... </details> -Theorem: Suppose that $V$ is finite-dimensional with a basis $\{ v_1, v_2, ..., v_n \}$. For any vectors $w_1, w_2, ... w_n$ in $W$, there exists *exactly* one linear transformation such that $T(v_i) = w_i$ for $i = 1, 2, ..., n$. +Theorem: Suppose that $V$ is finite-dimensional with a basis $\\{ v_1, v_2, ..., v_n \\}$. For any vectors $w_1, w_2, ... w_n$ in $W$, there exists *exactly* one linear transformation such that $T(v_i) = w_i$ for $i = 1, 2, ..., n$. <details> <summary>Proof</summary> ... </details> -## Linear Transformations as Matrices - -- Let $V, W$ be finite-dimensional vector spaces. -- Let $T, U : V → W$ be linear transformations from $V$ to $W$. -- Let $β$ and $γ$ be ordered bases of $V$ and $W$, respectively. -- Let $a ∈ F$ be a scalar. - -Definition: An **ordered basis** of a finite-dimensional vector space $V$ is, well, an ordered basis of $V$. We represent this with exactly the same notation as a standard unordered basis, but will call attention to it whenever necessary. -- For the vector space $F^n$ we call $\{ e_1, e_2, ..., e_n \}$ the **standard ordered basis** for $F^n$. -- For the vector space $P_n(F)$ we call $\{ 1, x, ..., x^n \}$ the **standard ordered basis** for $P_n(F)$. - -Definition: Let $a_1, a_2, ... a_n$ be the unique scalars such that $x = Σ_{i=1}^n a_i u_i$ for all $x ∈ V$. The **coordinate vector** of $x$ relative to $β$ is $(a_1, ..., a_n)$ (vert) and denoted $[x]_β$. - -Definition: The $m × n$ matrix $A$ defined by $A_{ij} = a_{ij}$ is called the **matrix representation of $T$ in the ordered bases $β$ and $γ$**, and denoted as $A = [T]_β^γ$. If $V = W$ and $β = γ$, we write $A = [T]_β$. +## Composition of Linear Transformations -Definition: Let $T, U : V → W$ be arbitrary functions. Let $a ∈ F$. We define $T + U : V → W$ as $(T + U)(x) = T(x) + U(x)$ for all $x ∈ V$, and $aT : V → W$ as $(aT)(x) = aT(x)$ for all $x ∈ V$. +Let $V$, $W$, and $Z$ be vector spaces. -Theorem: The set of all linear transformations (via our definitions of addition and scalar multiplication above) $V → W$ forms a vector space over $F$. +Theorem: The set of all linear transformations (via our definitions of addition and scalar multiplication above) $V → W$ forms a vector space over $F$. We denote this as $\mathcal{L}(V, W)$. If $V = W$, we write $\mathcal{L}(V)$. <details> <summary>Proof</summary> ... </details> -Definition: The vector space of all linear transformations $V → W$ is denoted by $\mathcal{L}(V, W)$. If $V = W$, we write $\mathcal{V}$. +Let $T, U : V → W$ be arbitrary functions. We define **addition** $T + U : V → W$ as $∀x ∈ V : (T + U)(x) = T(x) + U(x)$, and **scalar multiplication** $aT : V → W$ as $∀x ∈ V : (aT)(x) = aT(x)$ for all $a ∈ F$. -Theorem: $[T + U]_β^γ = [T]_β^γ + [U]_β^γ$ and $[aT]_β^γ = a[T]_β^γ$. +Theorem: Let $T : V → W$ and $U : W → Z$ be linear. Then their composition $UT : V → Z$ is linear. +<details> +<summary>Proof</summary> +Let $x,y ∈ V$ and $c ∈ F$. Then: + +$$UT(cx + y)$$ +$$= U(T(cx + y)) = U(cT(x) + T(y))$$ +$$= cU(T(x)) + U(T(y)) = c(UT)(x) + UT(y)$$ +</details> + +Theorem: Let $T, U_1, U_2 ∈ \mathcal{L}(V)$. Then: +- $T(U_1 + U_2) = TU_1 + TU_2$ and $(U_1 + U_2)T = U_1 T + U_2 T$ +- $T(U_1 U_2) = (TU_1) U_2$ +- $TI = IT = T$ +- $∀a ∈ F : a(U_1 U_2) = (aU_1) U_2 = U_1 (aU_2)$ <details> <summary>Proof</summary> ... +<!-- A more general result holds for linear transformations with domains unequal to their codomains, exercise 7 --> </details> -## Composition of Linear Transformations +## Linear Transformations as Matrices -Let $V$, $W$, and $Z$ be vector spaces. +- Let $V, W$ be finite-dimensional vector spaces. +- Let $T, U : V → W$ be linear transformations from $V$ to $W$. +- Let $β$ and $γ$ be ordered bases of $V$ and $W$, respectively. +- Let $a ∈ F$ be a scalar. -Theorem: Let $T : V → W$ and $U : W → Z$ be linear. Then their composition $UT : V → Z$ is linear. +An **ordered basis** of a finite-dimensional vector space $V$ is, well, an ordered basis of $V$. We represent this with exactly the same notation as a standard unordered basis, but will call attention to it whenever necessary. +- For the vector space $F^n$ we call $\\{ e_1, e_2, ..., e_n \\}$ the **standard ordered basis** for $F^n$. +- For the vector space $P_n(F)$ we call $\\{ 1, x, ..., x^n \\}$ the **standard ordered basis** for $P_n(F)$. + +Let $a_1, a_2, ... a_n$ be the unique scalars such that $x = Σ_{i=1}^n a_i u_i$ for all $x ∈ V$. The **coordinate vector** of $x$ relative to $β$ is $(a_1, ..., a_n)$ (vert) and denoted $[x]_β$. + +The $m × n$ matrix $A$ defined by $A_{ij} = a_{ij}$ is called the **matrix representation of $T$ in the ordered bases $β$ and $γ$**, and denoted as $A = [T]_β^γ$. If $V = W$ and $β = γ$, we write $A = [T]_β$. + +Theorem: $[T + U]_β^γ = [T]_β^γ + [U]_β^γ$ and $[aT]_β^γ = a[T]_β^γ$. <details> <summary>Proof</summary> -Let $x,y ∈ V$ and $c ∈ F$. Then: -$$UT(cx + y)$$ -$$= U(T(cx + y)) = U(cT(x) + T(y))$$ -$$= cU(T(x)) + U(T(y)) = c(UT)(x) + UT(y)$$ +... </details> -Definition: Let $T, -... +--- ## Invertibility and Isomorphism @@ -261,7 +268,7 @@ Let $V$ and $W$ be vector spaces. Let $T: U → V$ be a linear transformation. Let $I_V: V → V$ and $I_W: W → W$ denote the identity transformations within $V$ and $W$, respectively. -Definition: A function $U: W → V$ is an **inverse** of $T$ if $TU = I_W$ and $UT = I_V$. If $T$ has an inverse, then $T$ is **invertible**. +A function $U: W → V$ is an **inverse** of $T$ if $TU = I_W$ and $UT = I_V$. If $T$ has an inverse, then $T$ is **invertible**. Theorem: Consider a linear function $T: V → W$. - If $T$ is invertible, it has a *unique* inverse $T^{-1}$. @@ -278,7 +285,7 @@ Theorem: If $T$ is linear and invertible, $T^{-1}$ is linear and invertible. ... </details> -Definition: Let $A$ be a $n × n$ matrix. Then $A$ is **invertible** iff there exists an $n × n$ matrix $B$ such that $AB = BA = I$. +Let $A$ be a $n × n$ matrix. Then $A$ is **invertible** iff there exists an $n × n$ matrix $B$ such that $AB = BA = I$. Theorem: If $A$ is invertible, the matrix $B$ is unique, and denoted $A^{-1}$. <details> @@ -286,7 +293,7 @@ Theorem: If $A$ is invertible, the matrix $B$ is unique, and denoted $A^{-1}$. Suppose there existed another inverse matrix $C$. Then $C = CI = C(AB) = (CA)B = IB = B$. </details> -Definition: $V$ is **isomorphic** to $W$ if there exists an *invertible* linear transformation $T : V → W$ (an **isomorphism**). +$V$ is **isomorphic** to $W$ if there exists an *invertible* linear transformation $T : V → W$ (an **isomorphism**). Lemma: For finite-dimensional $V$ and $W$: If $T: V → W$ is invertible, then $dim(V) = dim(W)$. <details> |