Vector Spaces, Subspaces, Linear Independence, and Bases

Fundamental Notations and Geometric Vectors

• Standard Algebraic Fields:

  • $K$ denotes an arbitrary field.

  • $\mathbf{R}$ denotes the field of real numbers.

  • $\mathbf{C}$ denotes the field of complex numbers.

  • $\mathbf{Q}$ denotes the field of rational numbers.

• Geometric Definition in Physics:

  • In physical contexts, a vector is defined as a physical quantity that possesses both magnitude and direction.

  • Modern linear algebra generalizes this geometric concept to abstract mathematical structures where the elements may not admit a direct physical or visual representation.

Vectors in the Plane ($V_2(\mathbf{R})$)

• Points and Directed Line Segments:

  • A point in the plane is defined as an ordered pair $(a, b)$ of real numbers relative to a rectangular coordinate system.

  • The origin $O$ is the coordinate point $(0, 0)$.

  • A directed line segment from initial point $P$ to endpoint $Q$ is denoted by $\vec{PQ}$.

• Equivalence of Directed Line Segments:

  • Let $P = (p_1, p_2)$, $Q = (q_1, q_2)$, $R = (r_1, r_2)$, and $S = (s_1, s_2)$ be four points in the plane.

  • The two directed line segments $\vec{PQ}$ and $\vec{RS}$ are defined to be equivalent if:     $(q_1 - p_1, q_2 - p_2) = (s_1 - r_1, s_2 - r_2)$

  • This relation is an equivalence relation on the set of all directed line segments in the plane:

  • Reflexivity: $(q_1 - p_1, q_2 - p_2) = (q_1 - p_1, q_2 - p_2)$, so $\vec{PQ} \sim \vec{PQ}$.

  • Symmetry: If $(q_1 - p_1, q_2 - p_2) = (s_1 - r_1, s_2 - r_2)$, then $(s_1 - r_1, s_2 - r_2) = (q_1 - p_1, q_2 - p_2)$, so $\vec{RS} \sim \vec{PQ}$.

  • Transitivity: If $\vec{PQ} \sim \vec{RS}$ and $\vec{RS} \sim \vec{TU}$, where $T = (t_1, t_2)$ and $U = (u_1, u_2)$, then:       $(q_1 - p_1, q_2 - p_2) = (s_1 - r_1, s_2 - r_2) = (u_1 - t_1, u_2 - t_2)$       implying $\vec{PQ} \sim \vec{TU}$.

• Canonical Representation of Equivalence Classes:

  • Every directed line segment is equivalent to a unique directed line segment having the origin $O$ as its initial point.

  • Specifically, if $P = (p_1, p_2)$ and $Q = (q_1, q_2)$, then $\vec{PQ}$ is equivalent to $\vec{OR}$, where $R = (q_1 - p_1, q_2 - p_2)$.

  • Each equivalence class of directed line segments is uniquely represented by a directed line segment with its initial point at the origin $O$.

  • A vector $\vec{OP}$ in the plane is defined as the directed line segment with initial point $O$ and endpoint $P$.

  • Because a vector is completely determined by the coordinates of its endpoint, the vector $\vec{OP}$ is denoted by $(p_1, p_2)$.

• Operations in the Plane $V_2(\mathbf{R})$:

  • The set of all vectors in the plane is denoted by:     $V_2(\mathbf{R}) = \{(p_1, p_2) \mid p_1, p_2 \in \mathbf{R}\}$

  • Vector Addition (Parallelogram Rule): If $\vec{OP}$ and $\vec{OQ}$ are vectors, their sum is the vector $\vec{OR}$ corresponding to the diagonal of the parallelogram formed by $\vec{OP}$ and $\vec{OQ}$:     $\vec{OP} + \vec{OQ} = \vec{OR}$     $(p_1, p_2) + (q_1, q_2) = (p_1 + q_1, p_2 + q_2)$

  • Scalar Multiplication: Multiplication of a vector $\vec{OP}$ by a real scalar $\lambda$ produces a vector $\lambda \vec{OP} = \vec{OS}$ of length $|\lambda| \cdot OP$, pointing in the same direction as $\vec{OP}$ if $\lambda > 0$ and in the opposite direction if $\lambda < 0$:     $\lambda (p_1, p_2) = (\lambda p_1, \lambda p_2)$

Vectors in Space ($V_3(\mathbf{R})$) and $n$-Space ($V_n(K)$)

• Vectors in 3-Space ($V_3(\mathbf{R})$):

  • The set of all vectors in three-dimensional space is defined as:     $V_3(\mathbf{R}) = \{(\alpha, \beta, \gamma) \mid \alpha, \beta, \gamma \in \mathbf{R}\}$

  • Vector Addition:     $(\alpha, \beta, \gamma) + (\alpha', \beta', \gamma') = (\alpha + \alpha', \beta + \beta', \gamma + \gamma')$

  • Scalar Multiplication:     $\lambda (\alpha, \beta, \gamma) = (\lambda \alpha, \lambda \beta, \lambda \gamma)$

• Analytic Extension to $V_n(K)$ ($n \ge 4$):

  • For dimensions $n \ge 4$, geometric visualization is absent, necessitating a purely analytic definition over an arbitrary field $K$.

  • Definition of $V_n(K)$:     $V_n(K) = \{(\alpha_1, \alpha_2, \dots, \alpha_n) \mid \alpha_i \in K \ (i = 1, 2, \dots, n)\}$

  • Addition in $V_n(K)$:     $(\alpha_1, \alpha_2, \dots, \alpha_n) + (\beta_1, \beta_2, \dots, \beta_n) = (\alpha_1 + \beta_1, \alpha_2 + \beta_2, \dots, \alpha_n + \beta_n)$

  • Scalar Multiplication in $V_n(K)$:     $\lambda (\alpha_1, \alpha_2, \dots, \alpha_n) = (\lambda \alpha_1, \lambda \alpha_2, \dots, \lambda \alpha_n)$     for all $\lambda, \alpha_i, \beta_i \in K$.

• Theorem (Algebraic Laws of $V_n(K)$):

  • Let $\mathbf{u} = (\alpha_1, \alpha_2, \dots, \alpha_n)$, $\mathbf{v} = (\beta_1, \beta_2, \dots, \beta_n)$, and $\mathbf{w} = (\gamma_1, \gamma_2, \dots, \gamma_n)$ be arbitrary vectors in $V_n(K)$, and let $\alpha, \beta \in K$. The operations satisfy the following ten laws of algebra:

  • 1. Closure under vector addition: $\mathbf{u} + \mathbf{v} \in V_n(K)$

  • 2. Associativity of vector addition: $\mathbf{u} + (\mathbf{v} + \mathbf{w}) = (\mathbf{u} + \mathbf{v}) + \mathbf{w}$

  • 3. Commutativity of vector addition: $\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}$

  • 4. Existence of additive identity: There exists a zero element $\mathbf{0} = (0, 0, \dots, 0) \in V_n(K)$ such that $\mathbf{u} + \mathbf{0} = \mathbf{u}$.

  • 5. Existence of additive inverse: For each $\mathbf{u} \in V_n(K)$, there exists a vector $-\mathbf{u} = (-\alpha_1, -\alpha_2, \dots, -\alpha_n) \in V_n(K)$ such that $\mathbf{u} + (-\mathbf{u}) = \mathbf{0}$.

  • 6. Closure under scalar multiplication: $\alpha \mathbf{u} \in V_n(K)$

  • 7. Distributivity over scalar addition: $(\alpha + \beta)\mathbf{u} = \alpha \mathbf{u} + \beta \mathbf{u}$

  • 8. Distributivity over vector addition: $\alpha (\mathbf{u} + \mathbf{v}) = \alpha \mathbf{u} + \alpha \mathbf{v}$

  • 9. Associativity of scalar multiplication: $\alpha (\beta \mathbf{u}) = (\alpha \beta)\mathbf{u}$

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  10. Multiplication by identity scalar: $1\mathbf{u} = \mathbf{u}$, where $1$ is the multiplicative identity of $K$.

Axiomatic Definition and Verification of Vector Spaces

• Axiomatic Definition of a Vector Space:

  • A vector space consists of a non-empty set $V$ (whose elements are designated as vectors, denoted in lowercase bold) associated with two operations: vector addition ($+$) and scalar multiplication ($\cdot$) over an arbitrary field $K$, satisfying:

  • Part A: Closure Conditions:

  • 1. Closure under Vector Addition: For all $\mathbf{u}, \mathbf{v} \in V$, the sum $\mathbf{u} + \mathbf{v} = \mathbf{w}$ belongs to $V$.

  • 2. Closure under Scalar Multiplication: For all $\mathbf{u} \in V$ and $r \in K$, the product $r \cdot \mathbf{u} = \mathbf{v}$ belongs to $V$.

  • Part B: Operational Axioms:

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

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

  • 5. Associativity of Scalar Multiplication: $r \cdot (s \cdot \mathbf{u}) = (r \cdot s) \cdot \mathbf{u}$

  • 6. Existence of Zero Vector: A zero vector $\mathbf{0} \in V$ exists such that $\mathbf{u} + \mathbf{0} = \mathbf{u}$ for all $\mathbf{u} \in V$.

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

  • 8. Distributivity of Scalar Multiplication over Vector Addition: $r \cdot (\mathbf{u} + \mathbf{v}) = r \cdot \mathbf{u} + r \cdot \mathbf{v}$

  • 9. Distributivity of Scalar Addition over Scalar Multiplication: $(r + s) \cdot \mathbf{u} = r \cdot \mathbf{u} + s \cdot \mathbf{u}$

  • 
    

  10. Unitary Scalar Property: $1 \cdot \mathbf{u} = \mathbf{u}$ for all $\mathbf{u} \in V$.

• Scope and Generality of Vector Spaces:

  • Vectors need not be limited to standard coordinate lists from physics. Elements of a vector space may be:

  • Matrices of dimension $m \times n$

  • Continuous real-valued functions

  • Real or complex polynomials

  • Sets of solutions to differential equations

• Canonical Examples of Vector Spaces:

  • The field of real numbers $\mathbf{R}$ associated with standard addition and multiplication.

  • The field of complex numbers $\mathbf{C}$ associated with complex addition and scalar multiplication.

  • The set of all real polynomials $\mathbf{R}_n[x]$ under polynomial addition and scalar multiplication.

  • The set of all $n$-dimensional coordinate vectors $\mathbf{R}^n$ under standard componentwise operations.

  • The set of all $m \times n$ matrices under matrix addition and scalar multiplication.

  • The solution set of all functions $f(x)$ satisfying the differential equation $f(x) = f'(x)$.

• Complete Proof: Space of $2 \times 2$ Real Matrices:

  • Let $V$ be the set of all $2 \times 2$ matrices with real entries:

  • 1. Closure under Addition:       $\begin{pmatrix} a & b \\ c & d \end{pmatrix} + \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} a + e & b + f \\ c + g & d + h \end{pmatrix} \in V$

  • 2. Closure under Scalar Multiplication:       $r \cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} ra & rb \\ rc & rd \end{pmatrix} \in V$

  • 3. Commutativity:       $\begin{pmatrix} a & b \\ c & d \end{pmatrix} + \begin{pmatrix} e & f \\ g & h \end{pmatrix} = \begin{pmatrix} a + e & b + f \\ c + g & d + h \end{pmatrix} = \begin{pmatrix} e + a & f + b \\ g + c & h + d \end{pmatrix} = \begin{pmatrix} e & f \\ g & h \end{pmatrix} + \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

  • 4. Associativity of Addition:       $\left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} + \begin{pmatrix} e & f \\ g & h \end{pmatrix} \right) + \begin{pmatrix} i & j \\ k & l \end{pmatrix} = \begin{pmatrix} (a + e) + i & (b + f) + j \\ (c + g) + k & (d + h) + l \end{pmatrix}$       $= \begin{pmatrix} a + (e + i) & b + (f + j) \\ c + (g + k) & d + (h + l) \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix} + \left( \begin{pmatrix} e & f \\ g & h \end{pmatrix} + \begin{pmatrix} i & j \\ k & l \end{pmatrix} \right)$

  • 5. Associativity of Multiplication:       $r \left( s \begin{pmatrix} a & b \\ c & d \end{pmatrix} \right) = r \begin{pmatrix} sa & sb \\ sc & sd \end{pmatrix} = \begin{pmatrix} r(sa) & r(sb) \\ r(sc) & r(sd) \end{pmatrix} = \begin{pmatrix} (rs)a & (rs)b \\ (rs)c & (rs)d \end{pmatrix} = (rs) \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

  • 6. Zero Vector:       $\begin{pmatrix} a & b \\ c & d \end{pmatrix} + \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix} = \begin{pmatrix} a + 0 & b + 0 \\ c + 0 & d + 0 \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

  • 7. Negative Vector:       $\begin{pmatrix} a & b \\ c & d \end{pmatrix} + \begin{pmatrix} -a & -b \\ -c & -d \end{pmatrix} = \begin{pmatrix} a + (-a) & b + (-b) \\ c + (-c) & d + (-d) \end{pmatrix} = \begin{pmatrix} 0 & 0 \\ 0 & 0 \end{pmatrix}$

  • 8. Distributivity of Matrix Sums:       $r \left( \begin{pmatrix} a & b \\ c & d \end{pmatrix} + \begin{pmatrix} e & f \\ g & h \end{pmatrix} \right) = \begin{pmatrix} r(a + e) & r(b + f) \\ r(c + g) & r(d + h) \end{pmatrix} = \begin{pmatrix} ra + re & rb + rf \\ rc + rg & rd + rh \end{pmatrix}$       $= \begin{pmatrix} ra & rb \\ rc & rd \end{pmatrix} + \begin{pmatrix} re & rf \\ rg & rh \end{pmatrix} = r \begin{pmatrix} a & b \\ c & d \end{pmatrix} + r \begin{pmatrix} e & f \\ g & h \end{pmatrix}$

  • 9. Distributivity of Scalar Sums:       $(r + s) \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} (r + s)a & (r + s)b \\ (r + s)c & (r + s)d \end{pmatrix} = \begin{pmatrix} ra + sa & rb + sb \\ rc + sc & rd + sd \end{pmatrix}$       $= \begin{pmatrix} ra & rb \\ rc & rd \end{pmatrix} + \begin{pmatrix} sa & sb \\ sc & sd \end{pmatrix} = r \begin{pmatrix} a & b \\ c & d \end{pmatrix} + s \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

  • 
    

  10. Multiplication by Unit Scalar:       $1 \cdot \begin{pmatrix} a & b \\ c & d \end{pmatrix} = \begin{pmatrix} 1a & 1b \\ 1c & 1d \end{pmatrix} = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$

• Continuity and Polynomial Space Proofs:

  • Continuous Real Functions on $(-\infty, \infty)$:

  • If $f(x)$ and $g(x)$ are real continuous functions on $\mathbf{R}$ and $r \in \mathbf{R}$, then $(f + g)(x) = f(x) + g(x)$ is continuous on $\mathbf{R}$, and $(rf)(x) = r f(x)$ is continuous on $\mathbf{R}$.

  • Closure under addition and scalar multiplication is satisfied, and the remaining 8 axioms are inherited directly from the field properties of real numbers.

  • Real Polynomials with Degree $n \le 3$:

  • The sum of two polynomials of degree $\le 3$ has degree $\le 3$.

  • The product of a polynomial of degree $\le 3$ by a real scalar has degree $\le 3$.

  • Closure under both operations holds, and algebraic field properties ensure the remaining axioms are satisfied.

• Counterexamples (Sets that are NOT Vector Spaces):

  • Polynomials of Exact Degree $n = 4$:

  • Not closed under vector addition.

  • Counterexample: Consider $P(x) = -2x^4 + 3x^2 + 4x + 10$ and $Q(x) = 2x^4 + x^2 - 4x + 1$, both of degree 4.

  • Adding them yields:       $P(x) + Q(x) = (-2x^4 + 3x^2 + 4x + 10) + (2x^4 + x^2 - 4x + 1) = 4x^2 + 11$

  • The sum is of degree 2, not degree 4. Hence, the set fails closure under addition and is not a vector space.

  • Set of Integers ($\mathbf{Z}$) with Real Scalars ($\mathbf{R}$):

  • Not closed under scalar multiplication.

  • Counterexample: Let $x = -3 \in \mathbf{Z}$ and choose real scalar $r = \sqrt{3} \in \mathbf{R}$.

  • The product is $r \cdot x = -3\sqrt{3} \notin \mathbf{Z}$. Hence, the set is not a vector space.

Vector Subspaces

• Definition of a Subspace:

  • If $W$ is a subset of a vector space $V$, and $W$ is itself a vector space under the inherited operations of addition and scalar multiplication defined on $V$, then $W$ is called a subspace of $V$.

• Subspace Verification Criteria:

  • To demonstrate that a subset $W$ is a subspace of $V$, it is necessary and sufficient to establish:

  • 1. $W$ is a non-empty subset of $V$ ($W \subseteq V$).

  • 2. The zero vector $\mathbf{0}$ of $V$ belongs to $W$.

  • 3. Closure under addition: For any $\mathbf{u}, \mathbf{v} \in W$, $\mathbf{u} + \mathbf{v} \in W$.

  • 4. Closure under scalar multiplication: For any $\mathbf{u} \in W$ and scalar $r$, $r \cdot \mathbf{u} \in W$.

• Subspace Verification Examples:

  • Example 1: Vectors of the Form $(x, 0)$ in $\mathbf{R}^2$:

  • 1. $W = \{(x, 0) \mid x \in \mathbf{R}\} \subseteq \mathbf{R}^2$.

  • 2. The zero vector $(0, 0) \in W$ since the second component is 0.

  • 3. Closure under addition: $(x_1, 0) + (x_2, 0) = (x_1 + x_2, 0) \in W$.

  • 4. Closure under scalar multiplication: $r \cdot (x, 0) = (rx, 0) \in W$.

  • Therefore, $W$ is a subspace of $\mathbf{R}^2$.

  • Example 2: First Quadrant Vectors in $\mathbf{R}^2$:

  • Let $W = \{(x, y) \in \mathbf{R}^2 \mid x \ge 0, y \ge 0\}$.

  • While closed under addition, it is not closed under scalar multiplication.

  • Counterexample: Vector $\mathbf{u} = (3, 3) \in W$, but for scalar $r = -1$, $r\mathbf{u} = -1(3, 3) = (-3, -3) \notin W$.

  • Therefore, $W$ is not a subspace of $\mathbf{R}^2$.

• Example 3: Plane Passing Through the Origin in $\mathbf{R}^3$:

  • Let $W = \{(x, y, z) \in \mathbf{R}^3 \mid x + y + z = 0\}$.

  • 1. $W \subseteq \mathbf{R}^3$.

  • 2. $(0, 0, 0) \in W$ because $0 + 0 + 0 = 0$.

  • 3. Closure under addition: Let $\mathbf{u} = (x_1, y_1, z_1)$ and $\mathbf{v} = (x_2, y_2, z_2)$ be in $W$, so $x_1 + y_1 + z_1 = 0$ and $x_2 + y_2 + z_2 = 0$.     $\mathbf{u} + \mathbf{v} = (x_1 + x_2, y_1 + y_2, z_1 + z_2)$     $(x_1 + x_2) + (y_1 + y_2) + (z_1 + z_2) = (x_1 + y_1 + z_1) + (x_2 + y_2 + z_2) = 0 + 0 = 0$     Thus, $\mathbf{u} + \mathbf{v} \in W$.

  • 4. Closure under scalar multiplication: Let $r \in \mathbf{R}$. Then $r\mathbf{u} = (rx_1, ry_1, rz_1)$.     $rx_1 + ry_1 + rz_1 = r(x_1 + y_1 + z_1) = r \cdot 0 = 0$     Thus, $r\mathbf{u} \in W$.

  • Therefore, $W$ is a subspace of $\mathbf{R}^3$.

Linear Combinations of Vectors

• Definition of Linear Combination:

  • A vector $\mathbf{v}$ is called a linear combination of vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ if it can be expressed in the form:     $\mathbf{v} = r_1 \mathbf{u}_1 + r_2 \mathbf{u}_2 + \dots + r_n \mathbf{u}_n$     where $r_1, r_2, \dots, r_n$ are scalars.

• Example 1: Expressing a Vector as a Linear Combination:

  • Express vector $\mathbf{v} = \begin{pmatrix} 2 \\ 6 \\ 12 \end{pmatrix}$ in terms of $\mathbf{u}_1 = \begin{pmatrix} 1 \\ 3 \\ 0 \end{pmatrix}$ and $\mathbf{u}_2 = \begin{pmatrix} 0 \\ 0 \\ 4 \end{pmatrix}$.

  • Set up equation:     $\begin{pmatrix} 2 \\ 6 \\ 12 \end{pmatrix} = r_1 \begin{pmatrix} 1 \\ 3 \\ 0 \end{pmatrix} + r_2 \begin{pmatrix} 0 \\ 0 \\ 4 \end{pmatrix} = \begin{pmatrix} r_1 \\ 3r_1 \\ 4r_2 \end{pmatrix}$

  • Equating components gives:     $r_1 = 2$     $3r_1 = 6 \implies 3(2) = 6$     $4r_2 = 12 \implies r_2 = 3$

  • Hence:     $\begin{pmatrix} 2 \\ 6 \\ 12 \end{pmatrix} = 2 \begin{pmatrix} 1 \\ 3 \\ 0 \end{pmatrix} + 3 \begin{pmatrix} 0 \\ 0 \\ 4 \end{pmatrix}$

• Example 2: Determining Non-Representability:

  • Test whether $\mathbf{v} = \begin{pmatrix} 1 \\ 3 \end{pmatrix}$ can be represented by $\mathbf{u}_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$ and $\mathbf{u}_2 = \begin{pmatrix} 2 \\ 4 \end{pmatrix}$.

  • Set up system:     $\begin{pmatrix} 1 \\ 3 \end{pmatrix} = r_1 \begin{pmatrix} 1 \\ 2 \end{pmatrix} + r_2 \begin{pmatrix} 2 \\ 4 \end{pmatrix} = \begin{pmatrix} r_1 + 2r_2 \\ 2r_1 + 4r_2 \end{pmatrix}$

  • Yielding the equations:     $r_1 + 2r_2 = 1$     $2r_1 + 4r_2 = 3$

  • Multiply the first equation by $-2$:     $-2r_1 - 4r_2 = -2$     $2r_1 + 4r_2 = 3$

  • Adding both equations produces the contradiction $0 = 1$.

  • The system possesses no solution; hence, $\mathbf{v}$ cannot be expressed as a linear combination of $\mathbf{u}_1$ and $\mathbf{u}_2$.

• Example 3: Parameter Values via Matrix Determinants:

  • Determine values of $k$ such that $\mathbf{v} = \begin{pmatrix} -1 \\ 5 \end{pmatrix}$ is a linear combination of $\mathbf{u}_1 = \begin{pmatrix} -3 \\ 2 \end{pmatrix}$ and $\mathbf{u}_2 = \begin{pmatrix} k \\ 8 \end{pmatrix}$.

  • Set up system:     $\begin{pmatrix} -1 \\ 5 \end{pmatrix} = r_1 \begin{pmatrix} -3 \\ 2 \end{pmatrix} + r_2 \begin{pmatrix} k \\ 8 \end{pmatrix} = \begin{pmatrix} -3r_1 + kr_2 \\ 2r_1 + 8r_2 \end{pmatrix}$

  • Yielding the scalar equations:     $-3r_1 + kr_2 = -1$     $2r_1 + 8r_2 = 5$

  • For this system to have a unique solution for $r_1$ and $r_2$, the determinant of the coefficient matrix must be nonzero:     $\begin{vmatrix} -3 & k \\ 2 & 8 \end{vmatrix} = (-3)(8) - (2)(k) = -24 - 2k \ne 0$

• Solving the inequality:   $-2k \ne 24 \implies k \ne -12$

• Thus, $\mathbf{v}$ can be expressed as a linear combination for any value of $k \ne -12$.

Vector Span and Spanning Sets

• Definition of Span:

  • If $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ are vectors in a vector space $V$, the set $W$ of all linear combinations of these vectors is a subspace of $V$ called the span of the vectors:     $W = \text{Span}\{\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n\}$

  • The vectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$ are said to span $W$.

• Example 1: Vectors Spanning $\mathbf{R}^2$:

  • Show that $\mathbf{u}_1 = \begin{pmatrix} 1 \\ 5 \end{pmatrix}$ and $\mathbf{u}_2 = \begin{pmatrix} -1 \\ 2 \end{pmatrix}$ span $\mathbf{R}^2$.

  • An arbitrary vector in $\mathbf{R}^2$ is denoted by $\begin{pmatrix} a \\ b \end{pmatrix}$. Set up:     $\begin{pmatrix} a \\ b \end{pmatrix} = r_1 \begin{pmatrix} 1 \\ 5 \end{pmatrix} + r_2 \begin{pmatrix} -1 \\ 2 \end{pmatrix} = \begin{pmatrix} r_1 - r_2 \\ 5r_1 + 2r_2 \end{pmatrix}$

  • Multiply the first equation by 2:     $2r_1 - 2r_2 = 2a$     $5r_1 + 2r_2 = b$

  • Add both equations to eliminate $r_2$:     $7r_1 = 2a + b \implies r_1 = \frac{2a + b}{7}$

  • Substitute $r_1$ back into the first equation to solve for $r_2$:     $r_2 = r_1 - a = \frac{2a + b}{7} - a = \frac{b - 5a}{7}$

  • Because real scalars $r_1$ and $r_2$ exist for all $a, b \in \mathbf{R}$, the vectors span $\mathbf{R}^2$.

• Example 2: Vectors Failing to Span $\mathbf{R}^2$:

  • Test whether $\mathbf{u}_1 = \begin{pmatrix} 1 \\ -2 \end{pmatrix}$ and $\mathbf{u}_2 = \begin{pmatrix} 2 \\ -4 \end{pmatrix}$ span $\mathbf{R}^2$.

  • Set up equation:     $\begin{pmatrix} a \\ b \end{pmatrix} = r_1 \begin{pmatrix} 1 \\ -2 \end{pmatrix} + r_2 \begin{pmatrix} 2 \\ -4 \end{pmatrix} = \begin{pmatrix} r_1 + 2r_2 \\ -2r_1 - 4r_2 \end{pmatrix}$

  • Multiply the top equation by 2:     $2r_1 + 4r_2 = 2a$     $-2r_1 - 4r_2 = b$

  • Adding equations gives:     $0 = 2a + b$

  • Whenever $2a + b \ne 0$, no solution exists. Thus, the vectors do not span $\mathbf{R}^2$.

• Example 3: Standard Unit Vectors Spanning $\mathbf{R}^3$:

  • Let $\mathbf{u}_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, $\mathbf{u}_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$, and $\mathbf{u}_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$.

  • For any arbitrary vector $\begin{pmatrix} a \\ b \\ c \end{pmatrix} \in \mathbf{R}^3$:     $\begin{pmatrix} a \\ b \\ c \end{pmatrix} = r_1 \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix} + r_2 \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix} + r_3 \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix} = \begin{pmatrix} r_1 \\ r_2 \\ r_3 \end{pmatrix}$

  • Solutions exist directly as $r_1 = a$, $r_2 = b$, and $r_3 = c$, proving that the vectors span $\mathbf{R}^3$.

• Example 4: General Vectors Spanning $\mathbf{R}^3$:

  • Show that $\mathbf{u}_1 = \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix}$, $\mathbf{u}_2 = \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix}$, and $\mathbf{u}_3 = \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix}$ span $\mathbf{R}^3$.

  • Set up equation:     $\begin{pmatrix} a \\ b \\ c \end{pmatrix} = r_1 \begin{pmatrix} 1 \\ 2 \\ 0 \end{pmatrix} + r_2 \begin{pmatrix} 0 \\ 1 \\ -1 \end{pmatrix} + r_3 \begin{pmatrix} 0 \\ 2 \\ 1 \end{pmatrix} = \begin{pmatrix} r_1 \\ 2r_1 + r_2 + 2r_3 \\ -r_2 + r_3 \end{pmatrix}$

  • Solve the system:     $r_1 = a$     $r_2 + 2r_3 = b - 2a$     $-r_2 + r_3 = c$

  • Adding the second and third equations gives $3r_3 = b + c - 2a$, so:     $r_3 = \frac{b + c - 2a}{3}$

  • Substitute $r_3$ into $-r_2 + r_3 = c$ to solve for $r_2$:     $r_2 = r_3 - c = \frac{b + c - 2a}{3} - c = \frac{b - 2c - 2a}{3}$

  • Complete solution set:     $r_1 = a, \quad r_2 = \frac{b - 2c - 2a}{3}, \quad r_3 = \frac{b + c - 2a}{3}$

  • Because unique solutions exist for any real components $a, b, c$, the vectors span $\mathbf{R}^3$.

• Example 5: Vectors Failing to Span $\mathbf{R}^3$ via Row Echelon Form:

  • Test whether $\mathbf{u}_1 = \begin{pmatrix} -1 \\ 2 \\ 0 \end{pmatrix}$, $\mathbf{u}_2 = \begin{pmatrix} 0 \\ 2 \\ -1 \end{pmatrix}$, and $\mathbf{u}_3 = \begin{pmatrix} 1 \\ -4 \\ 1 \end{pmatrix}$ span $\mathbf{R}^3$.

  • Set up matrix system:     $\begin{pmatrix} -1 & 0 & 1 \\ 2 & 2 & -4 \\ 0 & -1 & 1 \end{pmatrix} \begin{pmatrix} r_1 \\ r_2 \\ r_3 \end{pmatrix} = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$

  • Augmented matrix:     $\begin{pmatrix} -1 & 0 & 1 & \mid & a \\ 2 & 2 & -4 & \mid & b \\ 0 & -1 & 1 & \mid & c \end{pmatrix}$

  • Interchange row 1 and row 2:     $\begin{pmatrix} 2 & 2 & -4 & \mid & b \\ -1 & 0 & 1 & \mid & a \\ 0 & -1 & 1 & \mid & c \end{pmatrix}$

  • Add row 1 to twice row 2 ($2R_2 + R_1 \to R_2$):     $\begin{pmatrix} 2 & 2 & -4 & \mid & b \\ 0 & 2 & -2 & \mid & 2a + b \\ 0 & -1 & 1 & \mid & c \end{pmatrix}$

  • Add row 2 to twice row 3 ($2R_3 + R_2 \to R_3$):     $\begin{pmatrix} 2 & 2 & -4 & \mid & b \\ 0 & 2 & -2 & \mid & 2a + b \\ 0 & 0 & 0 & \mid & 2a + b + 2c \end{pmatrix}$

  • The third row produces the equation $0 = 2a + b + 2c$.

  • The system has no solution whenever $2a + b + 2c \ne 0$. Therefore, these vectors do not span $\mathbf{R}^3$.