linear-algebra

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Linear Algebra (1)

All the notes precisely follows Linear Algebra Done Right by Sheldon Axler

Vector Spaces

Complex Numbers

Definition 1.1: Complex Numbers

  1. A Complex Number is an order pair \((a, b)\), where \(a, b \in \mathbb{R}\).
  2. Addition and multiplication is defined as: \[x + y = (a + b) + (c + d) = (a + c, b + d)\] \[xy = (a + b) (c + d) = (ac - bd, ad + bc)\]
  3. \(i = (0, 1)\) and \(i^2 = (0, 1) (0, 1) = -1\)
  4. Every complex number can be written as \[x = (a, b) = (a, 0) + (0, b) = (a, 0) + (b, 0)(0, 1) = a + bi\]
  5. The set of all complex numbers is denoted by \(\mathbb{C}\): \[\mathbb{C} = \{a + bi: a, b \in \mathbb{R}\}\]
  6. Addition and multiplication on \(\mathbb{C}\) are defined by \[(a + bi) + (c + di) = (a + c) + (b + d)i\] \[(a + bi) (c + di) = (ac - bd) + (ad + bc)i\]

\(a, c, b, d \in \mathbb{R}\)

If \(a \in \mathbb{R}\), we identify \(a + 0i\) with the real number \(a\). Thus we can think of \(\mathbb{R}\) as a subset of \(\mathbb{C}\).

Definition 1.2: Conjugate

If \(a, b\) are real and \(z = a + bi\), then the complex number \(\bar{z} = a - bi\) is called the conjugate of \(z\). The numbers \(a, b\) are the real part and the imaginary part of \(z\) respectively:

\[a = Re(z), \quad \quad b = Im(z)\]

Theorem 1.3A

If \(z\) and \(w\) are complex, then

  1. \(\overline{z + w} = \bar{z} + \bar{w}\)
  2. \(\overline{zw} = \bar{z}\bar{w}\)
  3. \(z + \bar{z} = 2Re(z),\quad z - \bar{z} = 2i Im(z)\)
  4. If \(z = a + bi\), \(z \bar{z} = a^2 + b^2\) is real and positive.
  5. \(|z| = \sqrt{z\bar{z}} = \sqrt{a^2 + b^2}\) is unique.
  6. \(|zw| = |z||w|\)

Theorem 1.3B: Schwarz Inequality

If \(a_1, ..., a_n\) and \(b_1, ..., b_n\) are complex numbers, then:

\[|\sum^n_{j=1} a_j \bar{b_j}|^2 \leq \sum^n_{j=1} |a_j|^2 \sum^{n}_{j=1} |b_j|^2\]

Property 1.4: Properties of Complex Arithmetic

  1. Commutativity:

    \[\alpha + \beta = \beta + \alpha, \;\; \alpha \beta = \beta \alpha, \quad \alpha, \beta \in \mathbb{C}\]

  2. Associativity:

    \[(\alpha + \beta) + \lambda = \alpha + (\beta + \lambda), \;\; (\alpha\beta)\lambda = \alpha (\beta\lambda), \quad \alpha, \beta, \lambda \in \mathbb{C}\]

  3. Identities:

    \[\lambda + 0 = \lambda, \;\; \lambda 1 = \lambda, \quad \forall \lambda \in \mathbb{C}\]

  4. Additive Inverse:

    For all \(\alpha \in \mathbb{C}\), there exists a unique \(\beta \in \mathbb{C}\) such that \(\alpha + \beta = 0\).

  5. Multiplicative Inverse:

    For all \(\alpha \in \mathbb{C}, \alpha \neq 0\), there exists a unique \(\beta \in \mathbb{C}\) such that \(\alpha\beta = 1\).

  6. Distributive Property:

    \[\lambda (\alpha + \beta) = \lambda \alpha + \lambda \beta, \quad \forall \lambda, \alpha, \beta \in \mathbb{C}\]

Definition 1.5: \(-\alpha\), Subtraction, \(\frac{1}{\alpha}\), Division

Let \(\alpha, \beta \in \mathbb{C}\):

  1. Let \(-\alpha\) denote the additive inverse of \(\alpha\). Thus, \(-\alpha\) is the unique complex number s.t: \[\alpha + (- \alpha) = 0\]
  2. Subtraction on \(\mathbb{C}\) is defined by: \[\beta - \alpha = \beta + (-\alpha)\]
  3. For \(\alpha \neq 0\), let \(\frac{1}{\alpha}\) denote the multiplicative inverse of \(\alpha\). Thus \(\frac{1}{\alpha}\) is the unique complex number s.t: \[\alpha (\frac{1}{\alpha}) = 1\]
  4. Division on \(\mathbb{C}\) is defined by: \[\frac{\beta}{\alpha} = \beta\frac{1}{\alpha}\]

Definition 1.7: \(\mathbb{F}\)

\(\mathbb{F}\) stands for either \(\mathbb{R}, \mathbb{C}\). Elements of \(\mathbb{F}\) are called scalars which is a fancy word for number.

Definition 1.8: List, Length

Suppose \(n\) is a nonnegative integer. A list of length \(n\) (\(n-tuple\)) is an ordered collection of \(n\) elements (can be numbers, other lists or more abstract entities) separated by commas and surrounded by parentheses. A list of length \(n\) looks like:

\[(x_1, ..., x_n)\]

Two lists are equal, if and only if they have the same length and the same elements in the same order.

Each list has a finite length that is a nonnegative integer. Thus an object with infinite length is not a list.

A list of length 0 is denoted as \(()\).

Definition 1.10: \(\mathbb{F}^n\)

\(\mathbb{F}^n\) is the set of all lists of length \(n\) of elements of \(\mathbb{F}\):

\[\mathbb{F}^{n} = \{(x_1, ..., x_n): x_j \in \mathbb{F}, \; \forall j=1, ..., n\}\]

For \((x_1, ..., x_n) \in \mathbb{F}^n\) and \(j \in \{1, ..., n\}\), we say that \(x_j\) is the \(j\)th coordinate of \((x_1, ..., x_n)\).

Definition 1.12: Addition in \(\mathbb{F}^n\)

Addition in \(\mathbb{F}^n\) is defined by adding corresponding coordinates:

\[(x_1, ...., x_n) + (y_1, ...., y_n) = (x_1 + y_1, ...., x_n + y_n)\]

Definition 1.13: Commutativity of Addition in \(\mathbb{F}^n\)

If \(x, y \in \mathbb{F}^n\), then \(x + y = y + x\)

If a single letter is used to denote an element of \(\mathbb{F}^n\), then the same letter with appropriate subscripts is often used when coordinates must be displayed.

Definition 1.14: 0

Let \(0\) denote the list of length \(n\) whose coordinates are all 0:

\[0 = (0, ..., 0)\]

Definition 1.16: Additive Inverse in \(\mathbb{F}^n\)

For \(x \in \mathbb{F}^n\), the additive inverse of \(x\), denoted \(-x\), is the vector \(-x \in \mathbb{F}^n\) such that

\[x + (-x) = 0\]

In order words, if \(x = (x_1, ...., x_n)\), then \(-x = (-x_1, ..., -x_n)\)

Definition 1.17: Scalar Multiplication in \(\mathbb{F}^n\)

The product of a number \(\lambda\) and a vector in \(\mathbb{F}^n\) is computed by multiplying each coordinate of the vector by \(\lambda\):

\[\lambda (x_1, ..., x_n) = (\lambda x_1, ...., \lambda x_n)\]

here \(\lambda \in \mathbb{F}, \;\ (x_1, ..., x_n) \in \mathbb{F}^n\)

Field

A field is a set containing at least two distinct elements called \(0, 1\), along with operations of addition and multiplication satisfying all the properties in Definition 1.3. Thus, \(\mathbb{R}, \mathbb{C}\) are fields.

Definition of Vector Space

Definition 1.18: Addition, Scalar Multiplication

  1. An addition on a set \(V\) is a function that assigns an element \(u + v \in V\) to each pair of elements \(u, v \in V\).
  2. A scalar multiplication on a set \(V\) is a function that assigns an element \(\lambda v \in V\) to each \(\lambda \in \mathbb{F}\) and each \(v \in V\).

Definition 1.19: Vector Space

A vector space is a set \(V\) along with an addition on \(V\) and a scalar multiplication on \(V\) such that the following properties hold:

  1. Commutativity: \[u + v = v + u, \; \forall u, v \in V\]

  2. Associativity: \[(u + v) + w = u + (v + w) \text{ and } (ab) v = a(bv), \; \forall u, v, w \in V, \; \forall a, b \in \mathbb{F}\]

  3. Additive Identity:

    There exists an element \(0 \in V\) such that \(v + 0 = v, \; \forall v \in V\)

  4. Additive Inverse: \[\forall v \in V, \exists \; w \in V \; s.t \; v + w = 0\]

  5. Multiplicative Identity: \[1v = v, \; \forall v \in V\]

  6. Distributive Properties: \[a(u + v) = au + av \text{ and } (a + b)v = av + bv, \; \forall a, b \in \mathbb{F}, \; \forall u, v \in \mathbb{F}\]

Definition 1.20: Vector, Point

Elements of a vector space are called vectors or points.

Since the scalar multiplication in a vector space depends on \(\mathbb{F}\), thus we need to be precise. We say that \(V\) is a vector space over \(\mathbb{F}\) (i.e \(\mathbb{R}^n\) is a vector space over \(\mathbb{R}\)).

Definition 1.21: Real Vector Space, Complex Vector Space

  1. A vector space over \(\mathbb{R}\) is called a real vector space.
  2. A vector space over \(\mathbb{C}\) is called a complex vector space.

Definition 1.23 \(\mathbb{F}^S\)

  1. If \(S\) is a set, then \(\mathbb{F}^S\) denotes the set of functions from \(S\) to \(\mathbb{F}\).
  2. For \(f, g \in \mathbb{F}^S\), the sum \(f + g \in \mathbb{F}^S\) is the function defined by: \[(f + g) (x) = f(x) + g(x), \; \forall x \in S\]
  3. For \(\lambda \in \mathbb{F}\) and \(f \in \mathbb{F}^S\), the product \(\lambda f \in \mathbb{F}^S\) is the function defined by: \[(\lambda f)(x) = \lambda f(x), \; \forall x \in S\]

We can think of \(\mathbb{F}^n\) as special case of \(\mathbb{F}^S\), because it can be represented as \(\mathbb{F}^{\{1, ...., n\}}\) which is the set of functions:

\[\{f: \{1, ..., n\} \rightarrow \mathbb{F}\}\]

and \(f(i)\) is the \(i\)th element of the list (i.e \((f(1) ,...., f(n)) \in \mathbb{F}^n\)). Since the sequence of numbers in \(\mathbb{F}\) is a function that maps from natural number to \(\mathbb{F}\).

Property 1.25: Unique Additive Identity

A vector space has a unique additive identity.

Property 1.26: Unique Additive Inverse

Every element in a vector space has a unique additive inverse.

Definition 1.27: \(-v, w - v\)

Let \(v, w \in V\), where \(V\) is a vector space. Then

  1. \(-v\) denotes the additive inverse of \(v\).
  2. \(w - v\) is defined to be \(w + (-v)\).

Property 1.29: The scalar 0 times a vector

\(0v = 0, \forall v \in V\), where \(V\) is a vector space over \(\mathbb{F}\).

Property 1.29: A scalar times a vector 0

\(a0 = 0, \forall a \in \mathbb{F}\).

Property 1.31: The number \(-1\) times a vector

\((-1)v = -v, \forall v \in V\), where \(V\) is a vector space over \(\mathbb{F}\)

Subspaces

Definition 1.32: Subspace (Linear Subspace)

A subset \(U\) of \(V\) is called a subspace of \(V\) if \(U\) is also a vector space (using the same addition adn scalar multiplication as on \(V\)).

Definition 1.34 Conditions for a Subspace

A subset \(U\) of \(V\) is a subspace of \(V\) if and only if \(U\) satisfies the following three conditions:

  1. Additive Identity: \[0 \in U\]
  2. Closed under Addition: \[u, w \in U \implies u + w \in U\]
  3. closed under scalar multiplication: \[a \in \mathbb{F}, u \in U \implies au \in U\]

Definition 1.36: Sum of Subsets

Suppose \(U_1, ...., U_m\) are subsets of \(V\). The sum of \(U_1, ..., U_m\), denoted \(U_1 + ... + U_m\), is the set of all possible sums of elements of \(U_1 ,...., U_m\):

\[U_1 + .... + U_m = \{u_1 + ... u_m: u_1 \in U_1, ..., u_m \in U_m\}\]

The union of subspaces is rarely a subspace which is why we usually work with sums rather than unions.

Definition 1.39: Sum of Subspaces is the Smallest Containing Subspace

Suppose \(U_1, ..., U_m\) are subspaces of \(V\). Then \(U_1 + .... + U_m\) is the smallest subspace of \(V\) containing \(U_1, ...., U_m\).

Definition 1.40: Direct Sum

Suppose \(U_1, ..., U_m\) are subspaces of \(V\):

  1. The sum \(U_1 + ... + U_m\) is called a direct sum if each element of the resulting set \(U_1 + ... + U_m\) can be written in only one way as a sum \(u_1 + ... + u_m\), where each \(u_j\) is in \(U_j\).
  2. If \(U_1 + .... + U_m\) is a direct sum, then \(U_1 \oplus ... \oplus U_m\) denotes \(U_1 + ... + U_m\), with the \(\oplus\) indicating the direct sum.

\[U_1 = \{(x, y, 0) \in \mathbb{F}^3: x, y \in \mathbb{F}\}\] \[U_2 = \{(0, 0, z) \in \mathbb{F}^3: x, y \in \mathbb{F}\}\] \[U_3 = \{(0, y, y) \in \mathbb{F}^3: x, y \in \mathbb{F}\}\]

Then, \(U_1 + U_2 + U_3 = \mathbb{F}^3\) is not a direct sum because the element \((0, 0, 0) \in \mathbb{F}^3\), can be written in two different ways using the sum:

\[(0, 1, 0) + (0, 0, 1) + (0, -1, -1) = (0, 0, 0)\] \[(0, 0, 0) + (0, 0, 0) + (0, 0, 0) = (0, 0, 0)\]

Definition 1.44 Condition for a Direct Sum

Suppose \(U_1, ...., U_m\) are subspaces of \(V\). Then \(U_1 + ... + U_m\) is a direct sum if and only if the only way to write \(0\) as a sum \(u_1 + ... + u_m\), where each \(u_j\) is in \(U_j\), is by taking each \(u_j\) equal to \(0\).

Definition 1.45 Direct Sum of Two Subspaces

Suppose \(U, W\) are subspaces of \(V\). Then \(U + W\) is a direct sum if and only if \(U \cap W = \{0\}\). The result only limit to two subspaces.

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