* Definition of a Vector Space: A set $V$ is a vector space over a field $F$ (typically $\mathbb{R}$ or $\mathbb{C}$) if for any vectors $\mathbf{u}, \mathbf{v}, \mathbf{w} \in V$ and scalars $c, d \in F$, the following axioms hold:

1. Closure under Addition: $\mathbf{u} + \mathbf{v} \in V$

2. Commutativity of Addition: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$

3. Associativity of Addition: $(\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})$

4. Existence of Zero Vector: There exists a zero vector $\mathbf{0} \in V$ such that $\mathbf{u} + \mathbf{0} = \mathbf{u}$

5. Existence of Additive Inverse: For each $\mathbf{u} \in V$, there exists $-\mathbf{u} \in V$ such that $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$

6. Closure under Scalar Multiplication: $c\mathbf{u} \in V$

7. Distributivity (Scalar over Vector Addition): $c(\mathbf{u} + \mathbf{v}) = c\mathbf{u} + c\mathbf{v}$

8. Distributivity (Vector over Scalar Addition): $(c + d)\mathbf{u} = c\mathbf{u} + d\mathbf{u}$

9. Associativity of Scalar Multiplication: $c(d\mathbf{u}) = (cd)\mathbf{u}$

10. Identity for Scalar Multiplication: $1\mathbf{u} = \mathbf{u}$

* $\mathbb{R}^n$: The set of all $n$-tuples of real numbers.

* $M_{m,n}$: The set of all $m \times n$ matrices with real entries.

* $P_n$: The set of all polynomials of degree less than or equal to $n$.

* $C[a,b]$: The set of all continuous real-valued functions defined on the interval $[a,b]$.

* $0\mathbf{u} = \mathbf{0}$

* $c\mathbf{0} = \mathbf{0}$

* $(-1)\mathbf{u} = -\mathbf{u}$

* If $c\mathbf{u} = \mathbf{0}$, then $c=0$ or $\mathbf{u}=\mathbf{0}$.

* Definition of a Subspace: A non-empty subset $W$ of a vector space $V$ is a subspace of $V$ if $W$ is a vector space under the operations of addition and scalar multiplication defined on $V$.

* Theorem for Testing Subspaces: A non-empty subset $W$ of a vector space $V$ is a subspace of $V$ if and only if it is closed under vector addition and scalar multiplication:

1. If $\mathbf{u}, \mathbf{v} \in W$, then $\mathbf{u} + \mathbf{v} \in W$.

2. If $\mathbf{u} \in W$ and $c$ is any scalar, then $c\mathbf{u} \in W$.

* The set of all $2 \times 2$ diagonal matrices is a subspace of $M_{2,2}$.

* The set of all polynomials of degree 2 or less ($P_2$) is a subspace of $P_3$.

* The set of all vectors $(x,y,z)$ in $\mathbb{R}^3$ such that $x+2y-z=0$ is a subspace of $\mathbb{R}^3$.

* Definition: A vector $\mathbf{v}$ in a vector space $V$ is a linear combination of vectors $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ in $V$ if $\mathbf{v}$ can be written in the form:

$\mathbf{v} = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_k\mathbf{v}_k$

where $c_1, c_2, \dots, c_k$ are scalars.

* Example: In $\mathbb{R}^3$, the vector $(1, -2, -5)$ is a linear combination of $\mathbf{v}_1 = (1, -2, -3)$ and $\mathbf{v}_2 = (2, -1, -4)$ because $(1, -2, -5) = -3(1, -2, -3) + 2(2, -1, -4)$.

* Definition: A set of vectors $S = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ in a vector space $V$ is a spanning set for $V$ if every vector in $V$ can be expressed as a linear combination of the vectors in $S$. In this case, we say that $V$ is spanned by $S$, or $S$ spans $V$, denoted as $V = \text{span}(S)$.

* Theorem: If $S = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ is a set of vectors in a vector space $V$, then $\text{span}(S)$ is a subspace of $V$.

* Example: The standard basis vectors $\{(1,0,0), (0,1,0), (0,0,1)\}$ span $\mathbb{R}^3$.

* Definition: A set of vectors $S = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k\}$ in a vector space $V$ is said to be linearly independent if the only solution to the vector equation:

$c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_k\mathbf{v}_k = \mathbf{0}$

is the trivial solution $c_1 = c_2 = \dots = c_k = 0$.

* Definition: If there are non-trivial solutions (i.e., at least one $c_i \neq 0$), then the set of vectors is linearly dependent.

* Test for Linear Independence: To check for linear independence, form a homogeneous system of linear equations from the vector equation and solve for the scalars $c_i$. If the only solution is $c_i=0$ for all $i$, the vectors are linearly independent.

* Wronskian for Functions: For a set of $n-1$ times differentiable functions $\{f_1(x), f_2(x), \dots, f_n(x)\}$, the Wronskian is defined as:

$W(f_1, \dots, f_n)(x) = \det \begin{pmatrix} f_1 & f_2 & \dots & f_n \\ f_1' & f_2' & \dots & f_n' \\ \vdots & \vdots & & \vdots \\ f_1^{(n-1)} & f_2^{(n-1)} & \dots & f_n^{(n-1)} \end{pmatrix}$

If $W(f_1, \dots, f_n)(x) \neq 0$ for some $x$ in an interval, then the functions are linearly independent on that interval.

* Definition of a Basis: A set of vectors $S = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ in a vector space $V$ is a basis for $V$ if the following two conditions are met:

1. $S$ spans $V$.

2. $S$ is linearly independent.

* Theorem: If $S = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ is a basis for a vector space $V$, then every vector $\mathbf{v} \in V$ can be expressed as a linear combination of the vectors in $S$ in exactly one way.

* Definition of Dimension: If a vector space $V$ has a basis consisting of $n$ vectors, then the dimension of $V$, denoted as $\dim(V)$, is $n$. If $V = \{\mathbf{0}\}$, then $\dim(V) = 0$.

* The standard basis for $\mathbb{R}^n$ is $\{\mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n\}$, so $\dim(\mathbb{R}^n) = n$.

* The standard basis for $P_n$ is $\{1, x, x^2, \dots, x^n\}$, so $\dim(P_n) = n+1$.

* The standard basis for $M_{m,n}$ consists of $mn$ matrices, each with a single 1 and the rest 0s, so $\dim(M_{m,n}) = mn$.

* Definition of Row Space: For an $m \times n$ matrix $A$, the row space of $A$, denoted as $\text{row}(A)$, is the subspace of $\mathbb{R}^n$ spanned by the row vectors of $A$.

* Definition of Column Space: For an $m \times n$ matrix $A$, the column space of $A$, denoted as $\text{col}(A)$, is the subspace of $\mathbb{R}^m$ spanned by the column vectors of $A$.

* Definition of Null Space: For an $m \times n$ matrix $A$, the null space of $A$, denoted as $\text{null}(A)$, is the set of all solutions to the homogeneous linear system $A\mathbf{x} = \mathbf{0}$.

$\text{null}(A) = \{\mathbf{x} \in \mathbb{R}^n \mid A\mathbf{x} = \mathbf{0}\}$

The null space is a subspace of $\mathbb{R}^n$.

* Row Space: A basis for the row space of $A$ consists of the non-zero row vectors of its row-echelon form (or reduced row-echelon form).

* Column Space: A basis for the column space of $A$ consists of the original column vectors of $A$ corresponding to the pivot columns in its row-echelon form.

* Null Space: A basis for the null space is found by solving $A\mathbf{x} = \mathbf{0}$ and expressing the solution in terms of free variables. The vectors associated with each free variable form a basis.

* Definition of Row Rank: The row rank of a matrix $A$ is the dimension of its row space, $\dim(\text{row}(A))$.

* Definition of Column Rank: The column rank of a matrix $A$ is the dimension of its column space, $\dim(\text{col}(A))$.

* Theorem: For any $m \times n$ matrix $A$, the row rank and column rank are equal. This common value is called the rank of $A$, denoted as $\text{rank}(A)$.

* Definition of Nullity: The nullity of a matrix $A$, denoted as $\text{nullity}(A)$, is the dimension of its null space, $\dim(\text{null}(A))$.

* Rank-Nullity Theorem: For an $m \times n$ matrix $A$, the sum of its rank and nullity is equal to the number of columns $n$:

$\text{rank}(A) + \text{nullity}(A) = n$

* Example: If a $3 \times 5$ matrix $A$ has a rank of 2, then its nullity is $5 - 2 = 3$.

* Coordinate Representation Relative to a Basis: If $B = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ is an ordered basis for a vector space $V$, and $\mathbf{v} \in V$ such that $\mathbf{v} = c_1\mathbf{v}_1 + c_2\mathbf{v}_2 + \dots + c_n\mathbf{v}_n$, then the coordinate vector of $\mathbf{v}$ relative to $B$ is:

$[\mathbf{v}]_B = \begin{pmatrix} c_1 \\ c_2 \\ \vdots \\ c_n \end{pmatrix}$

* Transition Matrix (Change of Basis Matrix): Let $B = \{\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_n\}$ and $B' = \{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$ be two bases for a vector space $V$. The transition matrix from $B'$ to $B$, denoted as $P_{B \leftarrow B'}$, is the matrix such that:

$[\mathbf{v}]_B = P_{B \leftarrow B'} [\mathbf{v}]_{B'}$

The columns of $P_{B \leftarrow B'}$ are the coordinate vectors of the basis vectors in $B'$ relative to the basis $B$:

$P_{B \leftarrow B'} = [[\mathbf{u}_1]_B \quad [\mathbf{u}_2]_B \quad \dots \quad [\mathbf{u}_n]_B]$

* Finding the Transition Matrix: To find $P_{B \leftarrow B'}$ for bases in $\mathbb{R}^n$, form the augmented matrix $[B \mid B']$. Row reduce this matrix to $[I \mid P_{B \leftarrow B'}]$.

* Inverse Relationship: The transition matrix from $B$ to $B'$ is the inverse of the transition matrix from $B'$ to $B$:

$P_{B' \leftarrow B} = (P_{B \leftarrow B'})^{-1}$

* Vector Space: A set $V$ equipped with vector addition and scalar multiplication satisfying ten axioms, allowing for operations similar to those with geometric vectors.

* Scalar: An element from the field $F$ (e.g., real numbers $\mathbb{R}$) used for scalar multiplication in a vector space.

* Zero Vector ($\mathbf{0}$): The unique additive identity element in a vector space, such that $\mathbf{v} + \mathbf{0} = \mathbf{v}$ for any vector $\mathbf{v}$.

* Additive Inverse ($-\mathbf{u}$): For every vector $\mathbf{u}$, there exists a unique vector $-\mathbf{u}$ such that $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$.

* Linear Combination: An expression of a vector as a sum of scalar multiples of other vectors, i.e., $c_1\mathbf{v}_1 + \dots + c_k\mathbf{v}_k$.

* Spanning Set: A set of vectors $S$ such that every vector in the vector space $V$ can be written as a linear combination of the vectors in $S$.

* Dimension ($\dim(V)$): The number of vectors in any basis for a vector space $V$.

* Row Space ($\text{row}(A)$): The subspace spanned by the row vectors of a matrix $A$.

* Column Space ($\text{col}(A)$): The subspace spanned by the column vectors of a matrix $A$.

* Null Space ($\text{null}(A)$): The set of all solutions to the homogeneous system $A\mathbf{x} = \mathbf{0}$.

* Rank ($\text{rank}(A)$): The dimension of the row space (or column space) of a matrix $A$. It equals the number of pivot columns in its row-echelon form.

* Nullity ($\text{nullity}(A)$): The dimension of the null space of a matrix $A$. It equals the number of free variables in the solution to $A\mathbf{x} = \mathbf{0}$.

* Coordinate Vector ($[\mathbf{v}]_B$): The column vector of scalars representing a vector $\mathbf{v}$ as a linear combination of the basis vectors in an ordered basis $B$.

* Transition Matrix ($P_{B \leftarrow B'}$): A matrix that transforms the coordinate vector of a vector relative to basis $B'$ to its coordinate vector relative to basis $B$.

* Verifying Vector Space Axioms: The document provides examples like $\mathbb{R}^n$, $M_{m,n}$, $P_n$, and $C[a,b]$ to illustrate how these sets satisfy the ten vector space axioms. For instance, it shows that the set of $2 \times 2$ matrices $M_{2,2}$ is a vector space by demonstrating closure under addition and scalar multiplication, and the existence of a zero matrix and additive inverses.

* Identifying Subspaces: Examples include showing that the set of all diagonal $2 \times 2$ matrices is a subspace of $M_{2,2}$ by checking closure under addition and scalar multiplication. Another example demonstrates that the set of vectors $(x,y,z)$ in $\mathbb{R}^3$ satisfying $x+2y-z=0$ forms a subspace.

* Expressing Vectors as Linear Combinations: A concrete example in $\mathbb{R}^3$ shows how to determine if a vector $(1, -2, -5)$ can be written as a linear combination of two other vectors $(1, -2, -3)$ and $(2, -1, -4)$ by setting up and solving a system of linear equations.

* Determining Spanning Sets: The document illustrates how to check if a set of vectors spans a given vector space, such as determining if $\{(1,1,1), (1,1,0), (1,0,0)\}$ spans $\mathbb{R}^3$ by verifying if any arbitrary vector $(x,y,z)$ can be written as their linear combination.

* Testing for Linear Independence: Examples involve setting up a homogeneous system $c_1\mathbf{v}_1 + \dots + c_k\mathbf{v}_k = \mathbf{0}$ and solving for $c_i$. For instance, it shows that $\{(1,2,3), (0,1,2), (-2,0,1)\}$ is linearly independent in $\mathbb{R}^3$ because the only solution is $c_1=c_2=c_3=0$. It also introduces the Wronskian for testing linear independence of functions.

* Finding a Basis and Dimension: The document provides examples of finding a basis for the solution space of a homogeneous system $A\mathbf{x}=\mathbf{0}$, which is the null space. This involves reducing the matrix to row-echelon form, identifying free variables, and expressing the solution in parametric vector form. The number of vectors in this basis gives the dimension of the null space (nullity).

* Bases for Row, Column, and Null Spaces: A detailed example with a $3 \times 5$ matrix $A$ demonstrates how to find bases for $\text{row}(A)$, $\text{col}(A)$, and $\text{null}(A)$. This involves reducing $A$ to row-echelon form to identify pivot rows (for row space basis) and pivot columns (for column space basis from original columns), and then solving $A\mathbf{x}=\mathbf{0}$ for the null space basis.

* Change of Basis: A comprehensive example in $\mathbb{R}^2$ shows how to find the coordinate vector of a vector relative to a non-standard basis $B'$. Then, it demonstrates how to find the transition matrix $P_{B \leftarrow B'}$ from $B'$ to the standard basis $B$, and how to use it to convert coordinates. It also shows how to find the transition matrix between two non-standard bases $B$ and $B'$ by forming the augmented matrix $[B \mid B']$ and row reducing it to $[I \mid P_{B \leftarrow B'}]$.

The "Xirius Vector Space 24 - MTH203" document provides a foundational and comprehensive exploration of vector spaces, a cornerstone concept in linear algebra. It begins by rigorously defining a vector space through ten axioms governing vector addition and scalar multiplication, emphasizing that these operations must be closed within the set and satisfy properties like commutativity, associativity, and the existence of identity and inverse elements. Key examples like $\mathbb{R}^n$ (n-tuples), $M_{m,n}$ (matrices), $P_n$ (polynomials), and $C[a,b]$ (continuous functions) are introduced to illustrate the broad applicability of the vector space concept beyond simple geometric vectors.

These concepts naturally lead to the definition of a basis – a set of vectors that is both linearly independent and spans the entire vector space. A basis represents the most efficient and non-redundant set of generators for a space. The number of vectors in any basis for a given vector space is unique and defines its dimension. The document provides standard bases for common vector spaces like $\mathbb{R}^n$, $P_n$, and $M_{m,n}$, and demonstrates how to find bases for more complex subspaces.

The discussion then shifts to the fundamental subspaces associated with a matrix $A$: the row space, column space, and null space. The row space is spanned by the rows of $A$, the column space by its columns, and the null space comprises all solutions to the homogeneous system $A\mathbf{x} = \mathbf{0}$. The document provides detailed procedures for finding bases for each of these spaces, typically involving row reduction of the matrix. This leads to the definitions of rank (the dimension of the row/column space) and nullity (the dimension of the null space). A pivotal result, the Rank-Nullity Theorem, states that for an $m \times n$ matrix $A$, $\text{rank}(A) + \text{nullity}(A) = n$, establishing a fundamental relationship between the dimensions of these subspaces and the number of columns in the matrix.

Finally, the document addresses the practical aspect of change of basis. It explains how to represent a vector using coordinate vectors relative to a specific ordered basis. The concept of a transition matrix is introduced as a tool to convert the coordinate vector of a vector from one basis to another. The document provides a clear method for constructing this transition matrix, particularly for bases in $\mathbb{R}^n$, by augmenting the basis matrices and performing row operations. This section is crucial for understanding how vector representations change when the underlying coordinate system is altered, a common operation in many applications.