Advanced Linear Algebra: Span, Linear Independence, and Real-World Applications

Geometrical Interpretation of Vector Scaling and Linear Combinations

• The concept of scaling vectors is foundational to linear algebra. If a vector $v_1$ is multiplied by a constant $c$, the resulting vector moves along a specific line in space.

• If the constant $c = 0$, the vector resides at the origin ($0, 0$).

• Scaling vectors allows movement exclusively along the line defined by that vector. For example, multiplying by positive numbers (2, 3, 4, 5) pulls the vector in the positive direction, while negative numbers pull it in the negative direction. It is impossible to move off this line using scaling alone.

• Graphically, this line represents the span of the vector. For a vector such as $v_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$, the equation of the line representing its span is $y = 2x$. In everyday algebra, this line passes through the origin with a slope of 2. No matter how the vector is scaled, it will never deviate from this line.

The Formal Definition of Span

• Suppose there is a set of vectors $S = \{v_1, v_2, \dots, v_k\}$. If these are vectors in $\mathbb{R}^n$, then the dimensions of each vector are $n$.

• Span Definition: The set of all linear combinations of $v_1, v_2, \dots, v_k$ is called the span of those vectors. This is denoted as $\text{span}\{v_1, v_2, \dots, v_k\}$ or $\text{span}(S)$.

• If $\text{span}(S) = \mathbb{R}^n$, then $S$ is referred to as the spanning set of $\mathbb{R}^n$.

• Example of Standard Spanning Sets:

  • In $\mathbb{R}^2$, the vectors $e_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ and $e_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ span the $xy$-coordinate system.

  • In $\mathbb{R}^3$, the standard unit vectors are $e_1 = \begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}$, $e_2 = \begin{pmatrix} 0 \\ 1 \\ 0 \end{pmatrix}$, and $e_3 = \begin{pmatrix} 0 \\ 0 \\ 1 \end{pmatrix}$. These vectors span all three dimensions ($x$, $y$, and $z$). Any point in $\mathbb{R}^3$, such as the point $\begin{pmatrix} -2 \\ 4 \\ 0 \end{pmatrix}$, can be represented as a linear combination of these unit vectors (e.g., $-2e_1 + 4e_2 + 0e_3$).

Linear Combinations and Systems Readiness

• A vector is a linear combination of other vectors if it can be written in the form $c_1v_1 + c_2v_2 + \dots + c_kv_k$.

• For a vector to be within the span of another set, a solution must exist for the system of equations. For example, representing the vector $\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}$ as a combination of two other vectors requires finding scalars $\alpha$ and $\beta$.

• Symbolic representation: If $\alpha = 3$ and $\beta = 2$, and these yield the target values in the vector, the answer is "Yes," the vector is a linear combination and thus lies within the span.

• Consistency and Inconsistency:

  • If a system of equations derived from vectors results in a contradiction like $0 = -2$, the system is inconsistent (e.g., the vectors may be parallel and never meet), meaning there is no solution.

  • If there are infinite solutions, the system is consistent.

Determining if Vectors Span $\mathbb{R}^2$

• To determine if two vectors, such as $\begin{pmatrix} 2 \\ -1 \end{pmatrix}$ and $\begin{pmatrix} 1 \\ 3 \end{pmatrix}$, span the entirety of $\mathbb{R}^2$, one must check if any arbitrary vector $\begin{pmatrix} a \\ b \end{pmatrix}$ can be represented by them.

• This leads to a system of equations that can be solved using row reduction. Swapping rows or multiplying by fractions (e.g., $\frac{1}{7}$) can help isolate the constants.

• In the case of these two vectors, the coefficients found are:

  • For the first vector: $\frac{6a + 2b}{7}$

  • For the second vector: $\frac{a + 2b}{7}$ (or similar derived constants such as $\frac{3a - b}{7}$ depending on row operations).

• Because solutions for the scalars exist for any choice of $a$ and $b$, these vectors do indeed span the $xy$-coordinate system.

Spanning Planes in $\mathbb{R}^3$

• To find the span of two vectors in $\mathbb{R}^3$, such as $s \begin{pmatrix} 1 \\ 0 \\ 3 \end{pmatrix} + t \begin{pmatrix} -1 \\ 1 \\ 3 \end{pmatrix}$, you set them equal to a general vector $\begin{pmatrix} x \\ y \\ z \end{pmatrix}$.

• Through row operations (e.g., $R_2 + R_1$ or multiplying by $-3$ and adding), one creates a row echelon form. The system will only have a solution if a specific condition is met, such as $z - 3x = 0$ or $3x - z = 0$.

• This shows that these vectors do not span the entirety of $\mathbb{R}^3$ but rather span a plane that passes through the origin. The general equation for such a surface is $ax + by + cz = 0$.

Linear Independence and Dependence

• Linear Dependence Definition: A set of vectors $\text{v}_1, \text{v}_2, \dots, \text{v}_k$ is linearly dependent if there are scalars $c_1, c_2, \dots, c_k$, at least one of which is non-zero, such that:   $c_1v_1 + c_2v_2 + \dots + c_kv_k = 0$

• Linear Independence: If the only solution to the above equation is the trivial solution where all scalars $c_i = 0$, then the vectors are linearly independent.

• Theorems on Dependence:

  • Two or more vectors are linearly dependent if and only if at least one vector can be expressed as a linear combination of the others. For example, if $v_1 = \begin{pmatrix} 1 \\ 2 \end{pmatrix}$, $v_2 = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$, and $v_3 = \begin{pmatrix} 4 \\ 6 \end{pmatrix}$, then $v_3$ is the sum of $v_1$ and $v_2$, making the set dependent.

  • Any set of vectors containing the zero vector is always linearly dependent. This is because the coefficient for the zero vector can be any non-zero number, while all other coefficients are zero, still resulting in a total of zero.

  • If you have $m$ vectors in $\mathbb{R}^n$ and m > n (more vectors than dimensions), the vectors are always linearly dependent.

Rank and Matrix Methods for Independence

• To check for independence, set up a matrix $A$ where the vectors are columns and the right-hand side is zero ($A\mathbf{x} = \mathbf{0}$).

• Use Row Reduced Echelon Form (RREF). If the only solution is zero for all variables, they are independent. If there are free variables or non-trivial solutions, they are dependent.

• Rank Theorem: A set of row vectors is linearly dependent if and only if \text{rank}(A) < m (where $m$ is the number of rows).

• Example: In a case where row reduction leads to a row of zeros (infinite solutions), the vectors are linearly dependent. If c3 is defined as a parameter $t$, then coefficients like $c_2 = 2t$ and $c_1 = -3t$ can be derived.

Application: Biological Strain Analysis

• Systems of equations can determine how many bacteria of different strains can coexist based on food consumption.

• Case Study: Three strains of bacteria consume three types of food (A, B, and C).

  • Units of A available: $2300$

  • Units of B available: $800$

  • Units of C available: $1500$

• The consumption rates for each strain form the coefficients of the matrix. For instance, if strain 1 eats 2 units of A, 1 of B, and 1 of C, and these are represented by variables $x_1, x_2, x_3$, the system is:   $2x_1 + 2x_2 + 4x_3 = 2300$   $1x_1 + 3x_2 + 0x_3 = 800$   $1x_1 + \dots = 1500$

• Solving via Row Echelon method yields: $x_1 = 100$, $x_2 = 350$, and $x_3 = 350$.

Application: Balancing Chemical Equations

• Chemistry uses linear systems to balance equations like $H_2 + O_2 \rightarrow H_2O$.

• Instead of trial and error, assign variables to the units of each molecule (e.g., $w, x, y, z$).

• Conservation of Mass: The number of atoms of each element must stay the same on both sides.

  • For Nitrogen (N): If the left side has $w$ and the right has $2y$, then $w = 2y$.

  • For Hydrogen (H): If the left has $3w$ and the right has $2z$, then $3w = 2z$.

  • For Oxygen (O): If the left has $2x$ and the right has $z$, then $2x = z$.

• This results in a system of equations that can be solved to find the smallest integer coefficients for balancing.

Application: Network Analysis and Flow

• Network analysis applies to traffic on streets, water in pipes, or electrical currents.

• Conservation of Flow: At any junction (node), the flow in must equal the flow out.

• Example Setup: At Junction A:

  • Flow In: $5 + 10 = 15$

  • Flow Out: $f_1 + f_4$

  • Equation: $f_1 + f_4 = 15$

• Similar equations are built for all junctions ($B, C, D$). For instance, if $f_2 + f_3 + 5$ enters Junction C and 30 leaves, then $f_2 + f_3 = 25$.

• These systems often result in infinite possibilities (parametric solutions) unless specific restrictions on maximum flow are provided.

Application: Electrical Networks

• Electrical circuits are managed using three primary principles:

  1. Ohm's Law: $E = I \times R$ (Voltage = Current $\times$ Resistance).

  2. Kirchhoff’s Current Law (KCL): The current entering a node must equal the current leaving (e.g., $I_1 + I_3 = I_2$).

  3. Kirchhoff’s Voltage Law (KVL): The total voltage drop around a closed circuit loop must equal the total voltage applied.

• Example Loop Analysis:

  • Loop 1: $4I_1 + I_2 = 8$

  • Loop 2: $4I_3 + I_2 = 16$

• Combining KCL and KVL creates a system of three formulas with three unknowns ($I_1, I_2, I_3$) which can be solved numerically using matrices.

Questions & Discussion: Test 1 Logistics

• Test Schedule: The test occurs on Wednesday from 10:10 AM to 12:10 PM (2 hours).

• Requirements:

  • Students must use the Lockdown Browser.

  • A working webcam and microphone are mandatory. The session is recorded, and the software flags suspicious activity (e.g., faces not being visible or too many "flags").

  • Scratch work must be written on physical paper and uploaded to the "Test 1 Scratch" section within 10–15 minutes after the test ends.

• Grading and Work:

  • The test is not multiple choice.

  • Partial credit is available, but no work means no credit, even if the final answer is correct.

  • Calculator use is allowed, but every step of the procedure must be shown.

• Submission:

  • Students must depart the Lockdown Browser to scan and upload their work.

  • Attachment vs. Upload: Students are asked to "attach" rather than just "upload" to allow the professor to use digital marking tools (red pen).

  • Email submissions will not be graded according to the syllabus.

Discussion on Technical Issues:

• Student Question: "Is the webcam going to be on our computer, or do we need another device?"

• Professor Response: Use the computer's webcam. Some professors ask for phones to see hands, but for this class, the computer's camera is sufficient.

• Student Question: "What app do we use to scan?"

• Professor Response: Any scanner app like Genius Scan or a printer's scanner works.

• Student Question: "Is the test similar to the homework?"

• Professor Response: Yes, the 9 to 10 questions are items you have seen before in the practice and homework.

Miscellaneous Aside

• During the break, the professor mentioned a personal note regarding burning through avocado oil and purchasing more at Costco.

• A brief interaction occurred with a student called "Mister Garcia" regarding administrative follow-up or assistance.