Linear Algebra

3.1 Vectors in 2-Space, 3-Space, and n-Space

Imron Rosyadi

Linear Algebra for ECE

Chapter 3: Vectors in Space

Introduction to Vectors

Linear algebra focuses on matrices and vectors.

This section extends your understanding of vectors from 2D/3D to \(n\)-space.

Today, we’re diving into vectors, fundamental building blocks in linear algebra. We’ll start with familiar geometric concepts in 2D and 3D, then generalize them to higher dimensions, which are crucial in many ECE applications. Think about how vectors represent physical quantities like force, velocity, or even signals.

Geometric Vectors

Engineers and physicists represent vectors using arrows.

• Direction: Specified by the arrowhead.
• Magnitude (Length): Specified by the length of the arrow.
• Initial Point: The tail of the arrow.
• Terminal Point: The tip of the arrow.

Geometric vectors are our intuitive starting point. They’re visual representations that help us understand the abstract concepts. In ECE, these could represent anything from a current flow direction to an electric field’s strength and direction. The key characteristics are direction and magnitude.

Vector Notation

A vector \(\mathbf{v}\) with initial point \(A\) and terminal point \(B\) is written as:

\[ \mathbf{v} = \overrightarrow{AB} \]

Boldface Notation: We denote vectors in boldface type, like \(\mathbf{a}, \mathbf{b}, \mathbf{v}, \mathbf{w}, \mathbf{x}\). Scalars are in lowercase italic type, like \(a, k, v, w, x\).

The notation \(\overrightarrow{AB}\) clearly indicates the start and end points. However, for general vectors, we use boldface letters. This distinction between vectors and scalars is important throughout linear algebra. Remember that a vector is defined by its magnitude and direction, not its specific location in space.

Equivalent Vectors and the Zero Vector

• Equivalent Vectors: Vectors with the same length and direction are considered equivalent (or equal). \[ \mathbf{v} = \mathbf{w} \] This means they are the “same vector” even if located differently.

• Zero Vector (\(\mathbf{0}\)): A vector whose initial and terminal points coincide, having zero length.

  • It has no natural direction, so we can assign any convenient direction.

The concept of equivalent vectors is crucial. It means we can move a vector around in space as long as we don’t change its length or direction. This is fundamental for operations like vector addition. The zero vector is like the number zero in arithmetic – it’s the additive identity.

Vector Addition: Geometric Rules

Parallelogram Rule: If \(\mathbf{v}\) and \(\mathbf{w}\) share an initial point, their sum \(\mathbf{v} + \mathbf{w}\) is the diagonal of the parallelogram formed by \(\mathbf{v}\) and \(\mathbf{w}\).

Triangle Rule (Tip-to-Tail): If the initial point of \(\mathbf{w}\) is at the terminal point of \(\mathbf{v}\), then \(\mathbf{v} + \mathbf{w}\) is the vector from the initial point of \(\mathbf{v}\) to the terminal point of \(\mathbf{w}\).

\[ \mathbf{v} + \mathbf{w} = \mathbf{w} + \mathbf{v} \tag{1} \]

Parallelogram Rule

Triangle Rule.

Commutativity.

These two rules are geometrically equivalent ways to visualize vector addition. The parallelogram rule is great when vectors start from the same point, like forces acting on an object. The triangle rule is useful for sequential movements or displacements. The commutative property (\(\mathbf{v} + \mathbf{w} = \mathbf{w} + \mathbf{v}\)) is clear from both rules.

Interactive Vector Addition (2D)

Adjust the components of \(\mathbf{v}\) and \(\mathbf{w}\) to see their sum.

viewof v1 = Inputs.range([-5, 5], {step: 0.1, label: "v1"});
viewof v2 = Inputs.range([-5, 5], {step: 0.1, label: "v2"});
viewof w1 = Inputs.range([-5, 5], {step: 0.1, label: "w1"});
viewof w2 = Inputs.range([-5, 5], {step: 0.1, label: "w2"});

This interactive plot helps visualize the parallelogram rule. You can manipulate the components of vectors \(\mathbf{v}\) and \(\mathbf{w}\) using the sliders. Observe how the resultant sum vector \(\mathbf{v} + \mathbf{w}\) changes dynamically. This is a great way to build intuition about vector addition before moving to more abstract definitions.

Vector Addition as Translation

Vector addition can also be seen as translating points.

• The terminal point of \(\mathbf{v} + \mathbf{w}\) is the point resulting from translating the terminal point of \(\mathbf{v}\) in the direction of \(\mathbf{w}\) by a distance equal to the length of \(\mathbf{w}\).
  • Equivalently: translating the terminal point of \(\mathbf{w}\) in the direction of \(\mathbf{v}\) by the length of \(\mathbf{v}\).

Translation of \(\mathbf{v}\) by \(\mathbf{w}\).

This perspective is particularly useful in computer graphics and robotics, where vectors often represent displacements. Imagine moving an object. The first vector is the initial movement, and the second vector is an additional movement from that new position. The final position is the sum of the two vectors from the original starting point.

Vector Subtraction

Subtraction is defined in terms of addition: \(a - b = a + (-b)\).

• Negative of a Vector (\(\mathbf{-v}\)): Has the same length as \(\mathbf{v}\) but is oppositely directed.
• Vector Subtraction (\(\mathbf{w} - \mathbf{v}\)): Defined as the sum \(\mathbf{w} + (-\mathbf{v})\). \[ \mathbf{w} - \mathbf{v} = \mathbf{w} + (-\mathbf{v}) \tag{2} \]

Geometrically, \(\mathbf{w} - \mathbf{v}\) is the vector from the terminal point of \(\mathbf{v}\) to the terminal point of \(\mathbf{w}\), when both start at the same initial point.

Vector subtraction is essentially adding the negative vector. The visual representation of \(\mathbf{w} - \mathbf{v}\) as the vector connecting the tips of \(\mathbf{v}\) and \(\mathbf{w}\) (when origins coincide) is very important for understanding relative positions or changes. In physics, if \(\mathbf{v}\) and \(\mathbf{w}\) are position vectors, \(\mathbf{w} - \mathbf{v}\) gives the displacement from the end of \(\mathbf{v}\) to the end of \(\mathbf{w}\).

Interactive Vector Subtraction (2D)

Adjust \(\mathbf{v}\) and \(\mathbf{w}\) to see their difference \(\mathbf{w} - \mathbf{v}\).

viewof sv1 = Inputs.range([-5, 5], {step: 0.1, label: "v1"});
viewof sv2 = Inputs.range([-5, 5], {step: 0.1, label: "v2"});
viewof sw1 = Inputs.range([-5, 5], {step: 0.1, label: "w1"});
viewof sw2 = Inputs.range([-5, 5], {step: 0.1, label: "w2"});

This interactive plot demonstrates vector subtraction. Notice how \(\mathbf{w} - \mathbf{v}\) connects the terminal point of \(\mathbf{v}\) to the terminal point of \(\mathbf{w}\) when both vectors originate from the same initial point. This is a direct visual representation of the difference between two position vectors.

Scalar Multiplication

Scalars change the length and/or reverse the direction of a vector.

• If \(k > 0\), \(k\mathbf{v}\) has the same direction as \(\mathbf{v}\) and length \(|k|\) times \(\mathbf{v}\).
• If \(k < 0\), \(k\mathbf{v}\) has the opposite direction of \(\mathbf{v}\) and length \(|k|\) times \(\mathbf{v}\).
• If \(k = 0\) or \(\mathbf{v} = \mathbf{0}\), then \(k\mathbf{v} = \mathbf{0}\).

From this, we see: \[ (-1)\mathbf{v} = -\mathbf{v} \tag{3} \]

Scalar multiplication is how we scale vectors. In ECE, scaling a voltage vector by a gain factor, or scaling a force vector, are common applications. The magnitude of the scalar \(k\) stretches or shrinks the vector, while its sign determines the direction.

Interactive Scalar Multiplication (2D)

Adjust the scalar \(k\) and vector \(\mathbf{v}\) to see the result \(k\mathbf{v}\).

viewof k_scalar = Inputs.range([-2, 2], {step: 0.1, label: "k"});
viewof sv_x = Inputs.range([-3, 3], {step: 0.1, label: "v_x"});
viewof sv_y = Inputs.range([-3, 3], {step: 0.1, label: "v_y"});

Here, you can experiment with scalar multiplication. Change k and observe how the vector kv scales and potentially reverses direction. What happens when k is 0? What about k=1 or k=-1? This interactive element helps solidify the geometric interpretation of scalar multiplication.

Parallel and Collinear Vectors

If one vector is a scalar multiple of another (\(\mathbf{w} = k\mathbf{v}\)), they are parallel.

• If they share a common initial point, they are also collinear.
• Translating a vector does not change it, so parallel and collinear mean the same thing for vectors.
• The zero vector \(\mathbf{0}\) is regarded as parallel to all vectors.

Parallel and collinear vectors.

The distinction between parallel and collinear can be confusing in everyday language. In linear algebra, when we say vectors are parallel, we mean they point in the same or opposite directions, regardless of their starting points. Collinear implies they lie on the same line. Since vectors can be moved, these terms become interchangeable.

Sums of Three or More Vectors

Vector addition is associative: \[ \mathbf{u} + (\mathbf{v} + \mathbf{w}) = (\mathbf{u} + \mathbf{v}) + \mathbf{w} \] This means the order of grouping doesn’t matter.

Tip-to-Tail Method: Place vectors sequentially (tip-to-tail). The sum is from the initial point of the first to the terminal point of the last.

Sum of three vectors (tip-to-tail).

The associative property simplifies multi-vector additions. The tip-to-tail method is a powerful visualization for any number of vectors. Think of navigating a robot: each vector is a movement command. The total displacement is the sum. The image on the right (3.1.9c) also shows the sum of three vectors as the main diagonal of a parallelepiped, extending the parallelogram rule to 3D.

Vectors in Coordinate Systems

Computations are simpler with coordinate systems.

• If a vector \(\mathbf{v}\) has its initial point at the origin, its components are the coordinates of its terminal point.
  • 2-space: \(\mathbf{v} = (v_1, v_2)\)
  • 3-space: \(\mathbf{v} = (v_1, v_2, v_3)\)
• The zero vector: \(\mathbf{0} = (0,0)\) in 2-space, \(\mathbf{0} = (0,0,0)\) in 3-space.

Vectors in coordinate systems.

Moving from geometric intuition to coordinate systems is where linear algebra becomes computational. This allows us to perform operations numerically, which is essential for ECE applications like signal processing or control systems. The components essentially give us a unique “address” for each vector.

Equivalent Vectors in Coordinates

Two vectors are equivalent if and only if their corresponding components are equal.

For \(\mathbf{v} = (v_1, v_2, v_3)\) and \(\mathbf{w} = (w_1, w_2, w_3)\) in 3-space: \[ \mathbf{v} = \mathbf{w} \quad \text{if and only if} \quad v_1 = w_1, \quad v_2 = w_2, \quad v_3 = w_3 \]

Note

Point vs. Vector: An ordered pair \((v_1, v_2)\) can represent a point (a location) or a vector (a displacement from the origin). The context determines the interpretation.

The ordered pair \((v_1, v_2)\) can represent a point or a vector.

This definition of equivalence is crucial for algebraic manipulation. It means we can compare vectors component by component. The callout highlights a common source of confusion: the dual interpretation of coordinates. In ECE, sometimes we care about the specific coordinates of a sensor, and sometimes about the vector representing its measurement.

Vectors Not at the Origin

If a vector’s initial point is not the origin, its components are found by subtracting the initial point’s coordinates from the terminal point’s.

• For \(\overrightarrow{P_1P_2}\) with \(P_1(x_1,y_1)\) and \(P_2(x_2,y_2)\): \[ \overrightarrow{P_1P_2} = (x_2 - x_1, y_2 - y_1) \tag{4} \]
• For \(\overrightarrow{P_1P_2}\) with \(P_1(x_1,y_1,z_1)\) and \(P_2(x_2,y_2,z_2)\): \[ \overrightarrow{P_1P_2} = (x_2 - x_1, y_2 - y_1, z_2 - z_1) \tag{5} \]

Vectors Not at the Origin

Vector \(\overrightarrow{P_1P_2}\) as difference of position vectors.

This formula effectively translates any vector so its tail is at the origin, allowing us to represent it by its terminal point’s coordinates. This is a standard way to represent displacement vectors in engineering. For example, the relative position of two components on a circuit board can be described this way.

EXAMPLE 1: Finding the Components of a Vector

Find the components of the vector \(\mathbf{v} = \overrightarrow{P_1P_2}\) with:

Initial point \(P_1(2, -1, 4)\)

Terminal point \(P_2(7, 5, -8)\)

\[ \mathbf{v} = (7 - 2, 5 - (-1), (-8) - 4) = (5, 6, -12) \]

Interactive Component Calculator

Enter coordinates to find vector components.

This example shows a direct application of the formula. The interactive calculator allows students to practice this skill with different coordinate sets. It’s a straightforward but fundamental calculation.

\(n\)-Space: Generalizing Dimensions

What if we need more than three dimensions?

• Ordered \(n\)-tuple: A sequence of \(n\) real numbers \((v_1, v_2, \ldots, v_n)\).
• \(n\)-Space (\(R^n\)): The set of all ordered \(n\)-tuples.

Tip

Think of \(n\)-tuples as: - Coordinates of a generalized point. - Components of a generalized vector. The choice depends on the problem context.

This is where linear algebra truly generalizes beyond our visual intuition. While we can’t see 4D or 11D space, the algebraic rules extend perfectly. This abstraction is powerful for modeling complex systems in ECE. For example, in signal processing, a signal sampled at \(N\) points can be represented as a vector in \(R^N\).

Applications of \(n\)-tuples in ECE and Beyond

• Experimental Data: \(n\) measurements form a vector \(\mathbf{y} = (y_1, \ldots, y_n)\) in \(R^n\).
  • Example: Sensor readings from an array.
• Storage/Warehousing: Distribution of items across \(n\) locations: \(\mathbf{x} = (x_1, \ldots, x_n)\).
  • Example: Inventory levels in a distributed system.
• Electrical Circuits: Input/output voltages for a chip.
  • Input \(\mathbf{v} \in R^4\), Output \(\mathbf{w} \in R^3\). Chip maps \(R^4 \to R^3\).

• Graphical Images: Pixel data (x, y, hue, saturation, brightness) as a 5-tuple.
  • \(\mathbf{v} = (x, y, h, s, b)\).
• Economics: Economic output of \(n\) sectors as \(\mathbf{s} = (s_1, \ldots, s_n)\).
• Mechanical Systems: State of particles (position, velocity, time).
  • \(\mathbf{v} = (x_1, \ldots, x_6, v_1, \ldots, v_6, t)\) in \(R^{13}\).

These examples illustrate the wide applicability of \(n\)-space. In ECE, think of state-space representations in control systems, feature vectors in machine learning, or even the parameters of a complex circuit model. The ability to represent diverse information as vectors in \(R^n\) is a cornerstone of modern engineering analysis.

Vector Equivalence in \(R^n\)

Vectors \(\mathbf{v} = (v_1, \ldots, v_n)\) and \(\mathbf{w} = (w_1, \ldots, w_n)\) in \(R^n\) are equivalent (equal) if:

\[ v_1 = w_1, \quad v_2 = w_2, \quad \ldots, \quad v_n = w_n \] We write this as \(\mathbf{v} = \mathbf{w}\).

EXAMPLE 2: Equality of Vectors

\[ (a, b, c, d) = (1, -4, 2, 7) \] if and only if \(a = 1, b = -4, c = 2,\) and \(d = 7\).

This definition of equality is a direct extension from 2D and 3D. It means that for two vectors to be considered the same, every single one of their corresponding components must match. This is fundamental for solving vector equations and verifying identities.

Algebraic Operations in \(R^n\) (Component-wise)

Operations on vectors in \(R^n\) are natural extensions of those in \(R^2\) and \(R^3\).

For \(\mathbf{v} = (v_1, v_2)\) and \(\mathbf{w} = (w_1, w_2)\) in \(R^2\): \[ \begin{array}{l} \mathbf{v} + \mathbf{w} = (v_1 + w_1, v_2 + w_2) \\ k\mathbf{v} = (k v_1, k v_2) \end{array} \tag{7} \] And similarly for subtraction and negative vectors: \[ -\mathbf{v} = (-v_1, -v_2) \tag{8} \] \[ \mathbf{w} - \mathbf{v} = (w_1 - v_1, w_2 - v_2) \tag{9} \]

Algebraic Operations in \(R^n\) (Component-wise)

Figure 3.1.13: Geometric interpretation of component-wise operations.

The beauty of coordinate systems is that vector operations become simple arithmetic on their components. This ‘component-wise’ approach is the workhorse of computational linear algebra. This slide shows how these operations are defined for 2D vectors, setting the stage for \(n\)-space.

Formal Definitions of Operations in \(R^n\)

If \(\mathbf{v} = (v_1, \ldots, v_n)\) and \(\mathbf{w} = (w_1, \ldots, w_n)\) are vectors in \(R^n\), and \(k\) is a scalar:

\[ \begin{array}{r l} \mathbf{v} + \mathbf{w} &= (v_1 + w_1, v_2 + w_2, \ldots, v_n + w_n)\\ k\mathbf{v} &= (k v_1, k v_2, \ldots, k v_n)\\ -\mathbf{v} &= (-v_1, -v_2, \ldots, -v_n)\\ \mathbf{w} - \mathbf{v} &= (w_1 - v_1, w_2 - v_2, \ldots, w_n - v_n) \end{array} \tag{13} \]

EXAMPLE 3: Algebraic Operations Using Components

If \(\mathbf{v} = (1, -3, 2)\) and \(\mathbf{w} = (4, 2, 1)\), then: \[ \begin{array}{r l} \mathbf{v} + \mathbf{w} &= (5, -1, 3) \\ 2\mathbf{v} &= (2, -6, 4) \\ -\mathbf{w} &= (-4, -2, -1) \\ \mathbf{v} - \mathbf{w} &= (-3, -5, 1) \end{array} \]

These are the formal definitions for \(n\)-space. It’s crucial to understand that all these operations are performed element-by-element on the vector components. The example illustrates these calculations with specific 3D vectors.

Interactive Vector Operations in \(R^3\)

Perform vector addition, subtraction, and scalar multiplication.

This Pyodide block lets you directly compute vector operations using NumPy. NumPy is a standard library in Python for numerical computing and handles vector/matrix operations very efficiently. You can modify the vectors v, w, and the scalar k in the code block and re-run to see different results. This is how these operations are typically performed in engineering software.

Properties of Vector Operations (Theorem 3.1.1)

If \(\mathbf{u},\mathbf{v},\) and \(\mathbf{w}\) are vectors in \(R^n\), and \(k\) and \(m\) are scalars:

1. \(\mathbf{u} + \mathbf{v} = \mathbf{v} + \mathbf{u}\) (Commutative Law for Addition)
2. \((\mathbf{u} + \mathbf{v}) + \mathbf{w} = \mathbf{u} + (\mathbf{v} + \mathbf{w})\) (Associative Law for Addition)
3. \(\mathbf{u} + \mathbf{0} = \mathbf{0} + \mathbf{u} = \mathbf{u}\) (Additive Identity)
4. \(\mathbf{u} + (-\mathbf{u}) = \mathbf{0}\) (Additive Inverse)
5. \(k(\mathbf{u} + \mathbf{v}) = k\mathbf{u} + k\mathbf{v}\) (Distributive Law)
6. \((k + m)\mathbf{u} = k\mathbf{u} + m\mathbf{u}\) (Distributive Law)
7. \(k(m\mathbf{u}) = (km)\mathbf{u}\) (Associative Law for Scalar Multiplication)
8. \(1\mathbf{u} = \mathbf{u}\) (Multiplicative Identity)

These properties are fundamental axioms that define a vector space. They are essential for manipulating vector equations and proving more complex theorems. They are direct extensions of properties you know from real number arithmetic.

Proof of Associative Law (b)

Let \(\mathbf{u} = (u_1, \ldots, u_n)\), \(\mathbf{v} = (v_1, \ldots, v_n)\), \(\mathbf{w} = (w_1, \ldots, w_n)\).

\[ \begin{array}{r l} (\mathbf{u} + \mathbf{v}) + \mathbf{w} &= \big((u_1,\ldots,u_n) + (v_1,\ldots,v_n)\big) + (w_1,\ldots,w_n)\\ &= (u_1 + v_1, \ldots, u_n + v_n) + (w_1,\ldots,w_n) & \text{[Vector addition]} \\ &= \big((u_1 + v_1) + w_1, \ldots, (u_n + v_n) + w_n\big) & \text{[Vector addition]} \\ &= \big(u_1 + (v_1 + w_1), \ldots, u_n + (v_n + w_n)\big) & \text{[Regroup real numbers]} \\ &= (u_1,\ldots,u_n) + (v_1 + w_1, \ldots, v_n + w_n) & \text{[Vector addition]} \\ &= \mathbf{u} + (\mathbf{v} + \mathbf{w}) \end{array} \]

This proof demonstrates how vector properties are derived directly from the properties of real numbers applied to their components. The key step is “Regroup real numbers,” where the associative property of scalar addition is used. This shows the consistency of our definitions.

Additional Vector Properties (Theorem 3.1.2)

If \(\mathbf{v}\) is a vector in \(R^n\) and \(k\) is a scalar:

1. \(0\mathbf{v} = \mathbf{0}\)
2. \(k\mathbf{0} = \mathbf{0}\)
3. \((-1)\mathbf{v} = -\mathbf{v}\)

These properties are also straightforward to prove using component-wise operations. They highlight the behavior of the zero scalar and the zero vector, and reinforce the definition of the negative of a vector.

Calculating Without Components

These theorems allow algebraic manipulation of vector equations without explicit component calculations.

Example: Solve \(\mathbf{x} + \mathbf{a} = \mathbf{b}\) for \(\mathbf{x}\).

\[ \begin{array}{r l r l} \mathbf{x} + \mathbf{a} &= \mathbf{b} & & \mathrm{[Given]}\\ (\mathbf{x} + \mathbf{a}) + (-\mathbf{a}) &= \mathbf{b} + (-\mathbf{a}) & & \mathrm{[Add~negative~of~a~to~both~sides]}\\ \mathbf{x} + (\mathbf{a} + (-\mathbf{a})) &= \mathbf{b} - \mathbf{a} & & \mathrm{[Part~}(b)\mathrm{~of~Thm~}3.1.1]\\ \mathbf{x} + \mathbf{0} &= \mathbf{b} - \mathbf{a} & & \mathrm{[Part~}(d)\mathrm{~of~Thm~}3.1.1]\\ \mathbf{x} &= \mathbf{b} - \mathbf{a} & & \mathrm{[Part~}(c)\mathrm{~of~Thm~}3.1.1] \end{array} \]

While it seems more cumbersome for \(R^n\), this abstract method is crucial when dealing with more general vector spaces where components might not be as straightforward (e.g., function spaces). It shows that the algebraic structure of vector operations is consistent and powerful.

Linear Combinations

A vector \(\mathbf{w}\) in \(R^n\) is a linear combination of vectors \(\mathbf{v}_1, \ldots, \mathbf{v}_r\) in \(R^n\) if it can be expressed as:

\[ \mathbf{w} = k_1\mathbf{v}_1 + k_2\mathbf{v}_2 + \dots + k_r\mathbf{v}_r \tag{14} \] where \(k_1, \ldots, k_r\) are scalars (coefficients).

Example: If \(\mathbf{v}_1, \mathbf{v}_2, \mathbf{v}_3\) are vectors:

\(\mathbf{u} = 2\mathbf{v}_1 + 3\mathbf{v}_2 + \mathbf{v}_3\)

\(\mathbf{w} = 7\mathbf{v}_1 - 6\mathbf{v}_2 + 8\mathbf{v}_3\)

Important

Linear combinations are central to many concepts in linear algebra, including span, basis, and transformations.

Linear combinations are one of the most important concepts in linear algebra. They allow us to build new vectors from existing ones. In ECE, think of superposition in circuits, where the total response is a linear combination of responses to individual inputs.

Interactive Linear Combination

Calculate a linear combination of two 2D vectors.

viewof lc_k1 = Inputs.range([-2, 2], {step: 0.1, label: "k1"});
viewof lc_k2 = Inputs.range([-2, 2], {step: 0.1, label: "k2"});
viewof lc_v1x = Inputs.range([-3, 3], {step: 0.1, label: "v1_x"});
viewof lc_v1y = Inputs.range([-3, 3], {step: 0.1, label: "v1_y"});
viewof lc_v2x = Inputs.range([-3, 3], {step: 0.1, label: "v2_x"});
viewof lc_v2y = Inputs.range([-3, 3], {step: 0.1, label: "v2_y"});

This interactive visualization demonstrates how a linear combination of two vectors creates a new vector. By adjusting the scalars k1, k2 and the vectors v1, v2, you can explore how the resultant vector k1*v1 + k2*v2 changes. Notice that the resultant vector always lies within the plane defined by v1 and v2 (if they are not collinear).

Alternative Notations for Vectors

Vectors can be written in several forms, depending on context and convenience.

1. Comma-delimited form: \[ \mathbf{v} = (v_1, v_2, \ldots, v_n) \tag{15} \]
2. Row-vector form: \[ \mathbf{v} = [v_1 \quad v_2 \quad \dots \quad v_n] \tag{16} \]
3. Column-vector form: \[ \mathbf{v} = \left[ \begin{array}{c}v_1 \\ v_2 \\ \vdots \\ v_n \end{array} \right] \tag{17} \]

While the comma-delimited form is common for general vector concepts, row and column vector forms are frequently used when vectors interact with matrices, like in matrix multiplication. In MATLAB or NumPy, column vectors are often preferred for certain linear algebra operations. It’s important to be comfortable with all notations.

Application: RGB Color Models

RGB color model: colors on computer monitors are created by adding percentages of Red (R), Green (G), and Blue (B).

• Primary colors as vectors in \(R^3\):
  • \(\mathbf{r} = (1,0,0)\) (pure red)
  • \(\mathbf{g} = (0,1,0)\) (pure green)
  • \(\mathbf{b} = (0,0,1)\) (pure blue)
• Any color \(\mathbf{c}\) in the RGB color cube is a linear combination: \[ \begin{array}{rl} \mathbf{c} &= k_1\mathbf{r} + k_2\mathbf{g} + k_3\mathbf{b}\\ &= k_1(1,0,0) + k_2(0,1,0) + k_3(0,0,1)\\ &= (k_1,k_2,k_3) \end{array} \] where \(0 \leq k_i \leq 1\).

Application: RGB Color Models

RGB Color Cube.

The RGB color model is a perfect real-world application of linear combinations and vectors in \(R^3\). Each color is a point (or vector) in a 3D space, where the axes represent red, green, and blue intensity. This is directly relevant to ECE students working with displays, image processing, or digital media.

Interactive RGB Color Mixer

Adjust the Red, Green, and Blue components (\(k_1, k_2, k_3\)) to mix colors.

viewof red_comp = Inputs.range([0, 1], {step: 0.01, label: "Red (k1)"});
viewof green_comp = Inputs.range([0, 1], {step: 0.01, label: "Green (k2)"});
viewof blue_comp = Inputs.range([0, 1], {step: 0.01, label: "Blue (k3)"});

This is a highly interactive demonstration of linear combinations in a practical ECE context. Adjust the Red, Green, and Blue sliders. Observe how the “Mixed Color” point moves within the RGB color cube and how its color changes. You can create a wide spectrum of colors by varying these three components. This directly shows how any color within the cube is a linear combination of the primary basis vectors.

Summary and Key Takeaways

• Geometric Vectors: Visualized as arrows, defined by magnitude and direction.
• Vector Operations: Addition, subtraction, and scalar multiplication have clear geometric and algebraic interpretations.
• Coordinate Systems: Allow for algebraic manipulation of vectors via components.
• \(n\)-Space: Generalizes vector concepts to higher dimensions, crucial for complex data.
• Linear Combinations: Fundamental for building new vectors and understanding vector relationships.
• Real-world Applications: Vectors are ubiquitous in ECE, from signal processing to computer graphics.

We’ve covered the foundational concepts of vectors, moving from intuitive geometric representations to formal algebraic definitions in \(n\)-space. Understanding these basics is essential for the rest of your linear algebra journey. Remember, linear algebra is the language of modern engineering and data science. Feel free to ask any questions!