Linear Algebra (Udacity)

1. Essence of Linear Algebra: Introduction

Common Issues in Learning Linear Algebra

Typically, students learn how to compute various operations, like matrix multiplication, determinants, cross products, or eigenvalues. However, they often don’t understand why these operations are defined as they are, or how they interrelate. For example:

• Matrix Multiplication: Why is it defined in this specific way?
• Determinants and Cross Products: What connects the determinant to the cross product?
• Eigenvalues: What do eigenvalues really represent?

Many students become proficient in the mechanics of these computations but have only a vague understanding of the geometric intuitions that underlie these concepts.

Importance of Geometric Understanding

There’s a significant difference between understanding linear algebra numerically and understanding it geometrically:

• Numeric Understanding: This is about carrying out calculations and applying formulas. It’s essential for practical computation.

• Geometric Understanding: This helps you judge which tools to use, why they work, and how to interpret the results. It’s crucial for problem-solving and gaining deep insight into the subject.

If you don’t develop a solid geometric understanding, the problems might not become apparent until you reach more advanced levels in your studies or career. In such situations, the way your professors or colleagues apply linear algebra might seem like magic, as they’ll quickly identify the right tools and approximate answers without needing to do heavy computations in their heads.

Analogy with Trigonometry

To illustrate this point, let’s consider an analogy with trigonometry. Imagine if, when first learning about the sine function, you were only shown the infinite series that defines it:

$$\sin (x)=x-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}+\dots$$

Your homework might involve computing sine by plugging numbers into this series and truncating it at a reasonable point. You might vaguely know this has something to do with triangles, but the connection isn’t clear. Later, in a physics course, when everyone else seems to immediately know how to apply sines and cosines, you’d likely feel overwhelmed. It might seem like others are somehow able to do these complex computations in their heads, making you feel slow or unintelligent.

This scenario isn’t too different from what happens with linear algebra. However, just like with trigonometry, linear algebra also has underlying visual intuitions. Fortunately, in linear algebra, the connection between these visual intuitions and the computations is often straightforward.

2. Vectors: A Unified Perspective

Vectors are fundamental building blocks in linear algebra, and understanding them from multiple perspectives can deepen your grasp of the subject. There are three distinct but related views of vectors that we will explore:

1. Physics Student Perspective:

   • Definition: Vectors are arrows pointing in space.
   • Key Characteristics: Each vector is defined by its length (magnitude) and the direction it points.
   • Dimensionality:
     • Vectors in a flat plane are two-dimensional (2D).
     • Vectors in the space we live in are three-dimensional (3D).
   • Important Concept: In physics, the exact position of the vector doesn’t matter as long as its magnitude and direction are unchanged. This means you can move the vector around, and it remains the same vector.

2. Computer Science Student Perspective:

   • Definition: Vectors are ordered lists of numbers.
   • Key Characteristics:
     • A vector represents a collection of values where the order matters.
     • For example, if analyzing house prices, you could represent each house as a vector: [square footage, price].
   • Dimensionality:
     • The number of elements in the list determines the dimensionality of the vector. A list with two elements (e.g., [1200, 300000]) is a two-dimensional vector.
   • Important Concept: In this context, a vector is like a structured data list, where each position in the list has a specific meaning.

3. Mathematician Perspective:

   • Definition: A vector is any object that can be added to another vector and multiplied by a scalar (a number), following certain rules.
   • Key Characteristics:
     • This view is more abstract and generalizes the previous two perspectives.
     • The focus is on operations like vector addition and scalar multiplication, which are fundamental in linear algebra.
   • Important Concept: This perspective is crucial for understanding vector spaces, where vectors aren’t just arrows or lists, but abstract entities with consistent algebraic properties.

Connecting the Perspectives

When we talk about vectors in linear algebra, it’s useful to blend these perspectives. Here’s how:

• Geometric Interpretation: Think of a vector as an arrow in a coordinate system, like the $xy$-plane. The vector is typically positioned with its tail at the origin (the point $(0,0)$).

  • Example: Consider a vector pointing from the origin to the point $(3,4)$ in the $xy$-plane. This vector can be thought of as the arrow starting at $(0,0)$ and ending at $(3,4)$.

• Numerical Representation: This vector can also be represented as the list [3, 4] in the computer science perspective, where the first number corresponds to the $x$-coordinate and the second to the $y$-coordinate.

Coordinate Systems and Their Importance

The coordinate system is the bridge between the geometric and numerical perspectives. It allows us to:

• Visualize Vectors: As arrows rooted at the origin in a plane or space.
• Translate Vectors to Lists: By considering the coordinates of the arrow’s tip as an ordered list.

Summary

Understanding vectors involves shifting between these perspectives:

1. Physics: Think of vectors as arrows with direction and magnitude.
2. Computer Science: Think of vectors as ordered lists of numbers.
3. Mathematics: Think of vectors as objects that can be added and scaled, with a focus on operations and properties.

By solidifying these perspectives, you’ll build a strong foundation for studying linear algebra, where vectors play a crucial role in almost every concept.

3. Vectors in Two and Three Dimensions

When working with vectors, it’s essential to understand how they are represented in both two-dimensional (2D) and three-dimensional (3D) spaces. Let’s break this down step by step:

1. Two-Dimensional Space (2D)

• Coordinate System Setup:

  • X-Axis: The horizontal line.
  • Y-Axis: The vertical line.
  • Origin: The point where the $x$-axis and $y$-axis intersect, denoted as $(0,0)$. This is the “root” of all vectors in this space.

• Grid and Tick Marks:

  • Choose an arbitrary length to represent one unit, and mark off distances along both axes. These tick marks are used to measure distances along the $x$-axis and $y$-axis.
  • When representing the entire 2D space, these tick marks can be extended into grid lines, creating a visual guide for positioning vectors.

• Vector Representation:

  • A vector in 2D is represented by a pair of numbers, $[x,y]$, which gives the “instructions” for how to move from the origin to the tip of the vector.
    • $x$: Indicates how far to move along the $x$-axis.
      • Positive $x$ means moving right.
      • Negative $x$ means moving left.
    • $y$: Indicates how far to move along the $y$-axis after moving along the $x$-axis.
      • Positive $y$ means moving up.
      • Negative $y$ means moving down.

• Notation:

  • To distinguish vectors from points, vectors are often written vertically with square brackets: $$\left[\begin{array}{c}x\\ y\end{array}\right]$$
  • This notation highlights the vector’s components along each axis.

• One-to-One Correspondence:

  • Each pair of numbers corresponds to exactly one vector in 2D space.
  • Conversely, every vector in 2D space is associated with one unique pair of numbers.

2. Three-Dimensional Space (3D)

• Extending to 3D:

  • Z-Axis: Add a third axis, the $z$-axis, which is perpendicular to both the $x$-axis and $y$-axis. This axis adds depth to our space.
  • The origin in 3D space is still where all three axes intersect, denoted as $(0,0,0)$.

• Vector Representation:

  • A vector in 3D is represented by a triplet of numbers, $[x,y,z]$, which gives the instructions for how to move from the origin to the tip of the vector.
    • $x$: How far to move along the $x$-axis.
    • $y$: How far to move parallel to the $y$-axis.
    • $z$: How far to move parallel to the $z$-axis.

• Notation:

  • In 3D, vectors are typically written as: $$\left[\begin{array}{c}x\\ y\\ z\end{array}\right]$$
  • This clearly shows the vector’s components along the $x$, $y$, and $z$ axes.

• One-to-One Correspondence:

  • Just like in 2D, each triplet of numbers corresponds to exactly one vector in 3D space.
  • Every vector in 3D space is associated with one unique triplet of numbers.

Summary

• In 2D: Vectors are represented by pairs of numbers, $[x,y]$, with movements along the $x$ and $y$ axes.
• In 3D: Vectors are represented by triplets of numbers, $[x,y,z]$, with movements along the $x$, $y$, and $z$ axes.

These concepts form the foundation for understanding more complex operations in linear algebra, as vectors are the primary objects you’ll be working with across various dimensions.

4. Vector Addition and Scalar Multiplication: Core Operations in Linear Algebra

In linear algebra, two fundamental operations are central to everything: vector addition and scalar multiplication. These operations are the building blocks for more complex concepts and are essential to understand.

1. Vector Addition

• Geometric Interpretation:

  • Let’s say we have two vectors:
    • The first vector points up and slightly to the right.
    • The second vector points to the right and slightly down.
  • To add these vectors:
    • Position the tail of the second vector at the tip of the first vector.
    • Draw a new vector from the tail of the first vector to the tip of the second vector in its new position.
    • This new vector represents the sum of the two vectors.

• Why This Works:

  • Think of each vector as a movement in space. The sum of two vectors represents the combined movement of first following the direction of one vector, then following the direction of the other.
  • This concept is similar to adding numbers on a number line. For instance, adding 2 and 5 can be thought of as moving two steps to the right and then five more steps to the right, resulting in a total movement of seven steps.

• Numerical Example:

  • Consider two vectors with coordinates:
    • ${\mathbf{v}}_1=\left[\begin{array}{c}1\\ 2\end{array}\right]$
    • ${\mathbf{v}}_2=\left[\begin{array}{c}3\\ -1\end{array}\right]$
  • To add these vectors, add their corresponding components: $${\mathbf{v}}_1+{\mathbf{v}}_2=\left[\begin{array}{c}1\\ 2\end{array}\right]+\left[\begin{array}{c}3\\ -1\end{array}\right]=\left[\begin{array}{c}1+3\\ 2+(-1)\end{array}\right]=\left[\begin{array}{c}4\\ 1\end{array}\right]$$
  • This process is straightforward: add the $x$-components together and the $y$-components together to get the resulting vector.

2. Scalar Multiplication

• Geometric Interpretation:

  • Scalar multiplication involves multiplying a vector by a number (called a scalar).
  • Examples:
    • Multiplying by 2 doubles the length of the vector.
    • Multiplying by $\frac{1}{3}$ shrinks the vector to one-third of its original length.
    • Multiplying by a negative number like $-1.8$ flips the vector’s direction and then stretches it by a factor of 1.8.

• Concept of Scaling:

  • The process of stretching, shrinking, or reversing the direction of a vector is known as scaling.
  • The number (scalar) used to scale the vector is called a scalar. In linear algebra, the terms “scalar” and “number” are often used interchangeably because scalars typically scale vectors.

• Numerical Example:

  • Consider a vector: $$\mathbf{v}=\left[\begin{array}{c}2\\ -3\end{array}\right]$$
  • If we multiply this vector by a scalar, say 3: $$3\times \mathbf{v}=3\times \left[\begin{array}{c}2\\ -3\end{array}\right]=\left[\begin{array}{c}3\times 2\\ 3\times -3\end{array}\right]=\left[\begin{array}{c}6\\ -9\end{array}\right]$$
  • This operation scales the vector by a factor of 3, stretching it while keeping the same direction.

Summary

• Vector Addition: Combine two vectors by adding their corresponding components, which geometrically represents combining two movements into one.
• Scalar Multiplication: Scale a vector by multiplying each of its components by a scalar, which stretches, shrinks, or flips the vector.

These operations are fundamental and will recur throughout your study of linear algebra. They lay the groundwork for more advanced topics such as span, basis, and linear dependence, which build directly on these basic operations.

5. Linear Combinations in Two-Dimensional Space

Key Concepts:

1. Vector Coordinates:

   • A vector in a 2D space can be described by a pair of numbers, like $(3,-2)$.
   • These numbers, or coordinates, are scalars that describe how much you stretch or shrink certain vectors.

2. Basis Vectors:

   • In the standard Cartesian coordinate system, two special vectors are used:
     • $\hat{i}$: A unit vector pointing to the right (in the x-direction).
     • $\hat{j}$: A unit vector pointing straight up (in the y-direction).
   • These vectors $\hat{i}$ and $\hat{j}$ form the basis of the coordinate system.

3. Scaling Vectors:

   • The x-coordinate scales the vector $\hat{i}$ and the y-coordinate scales the vector $\hat{j}$.
   • For example, in the vector $(3,-2)$, $3$ scales $\hat{i}$ (stretching it three times), and $-2$ scales $\hat{j}$ (flipping and stretching it by a factor of two).

4. Linear Combination:

   • A linear combination of two vectors involves scaling them by scalars and then adding them.
   • Example: Given vectors $\mathbf{a}$ and $\mathbf{b}$, a linear combination could be $c_1\mathbf{a}+c_2\mathbf{b}$, where $c_1$ and $c_2$ are scalars.

5. Span of Vectors:

   • The span of two vectors is the set of all possible vectors you can get by taking linear combinations of those vectors.
   • If the vectors are not aligned, their span covers the entire 2D plane.
   • If they are aligned (collinear), their span is a line through the origin.
   • If both vectors are zero, the span is just the origin point.

Visualization and Understanding:

• Fixed Scalars: If you fix one scalar (say $c_1$) and let the other vary (say $c_2$), the tip of the resulting vector will trace a line in 2D space.
• Varying Scalars: If both scalars are allowed to vary, the tips of the resulting vectors will cover an area in 2D space:
  • If the vectors are not collinear, this area is the entire plane.
  • If they are collinear, the area is just a line.

Example:

Consider vectors $\mathbf{u}=(2,3)$ and $\mathbf{v}=(1,1)$:

• A linear combination could be written as: $$\mathbf{w}=c_1\mathbf{u}+c_2\mathbf{v}=c_1\left(\begin{array}{c}2\\ 3\end{array}\right)+c_2\left(\begin{array}{c}1\\ 1\end{array}\right)=\left(\begin{array}{c}2c_1+c_2\\ 3c_1+c_2\end{array}\right)$$
• Depending on $c_1$ and $c_2$, $\mathbf{w}$ can represent any vector in the 2D plane if $\mathbf{u}$ and $\mathbf{v}$ are not collinear.

Mathematical Representation:

• Matrix Form: A linear combination can be expressed as a matrix equation: $$\mathbf{w}=c_1\mathbf{u}+c_2\mathbf{v}=\left(\begin{array}{cc}\mathbf{u}& \mathbf{v}\end{array}\right)\left(\begin{array}{c}{c}_1\\ c_2\end{array}\right)$$ where $\mathbf{u}$ and $\mathbf{v}$ are column vectors.

This framework of using basis vectors, scaling, and adding them through linear combinations is fundamental in linear algebra and helps build an understanding of vector spaces, transformations, and much more.

6. Span and Linear Independence in Higher Dimensions

Key Concepts:

1. Vectors as Points:

   • Vectors can be represented as points in space, where the point is at the tip of the vector.
   • This representation is particularly useful when dealing with multiple vectors, as it simplifies the visualization by focusing on the tip rather than the entire arrow.

2. Span in 2D Space:

   • The span of two vectors in 2D space is the set of all possible linear combinations of those two vectors.
   • If the vectors are not collinear, their span covers the entire plane. If they are collinear, the span is just a line.

3. Span in 3D Space:

   • In 3D space, the span of two non-parallel vectors forms a flat plane (a 2D sheet) that cuts through the origin.
   • Adding a third vector can either keep the span within the plane (if the third vector lies on the plane) or extend the span to cover the entire 3D space.

3D Span Visualization:

• Two Vectors: Imagine two vectors $\mathbf{u}$ and $\mathbf{v}$ in 3D space. Their span forms a flat sheet. If you vary the scalars that scale $\mathbf{u}$ and $\mathbf{v}$, the tip of the resulting vector traces out this sheet.
• Adding a Third Vector: Introduce a third vector $\mathbf{w}$:
  • If $\mathbf{w}$ lies in the same plane as $\mathbf{u}$ and $\mathbf{v}$**, the span remains a plane.
  • If $\mathbf{w}$ does not lie in the plane, the span now covers the entire 3D space.

Linear Independence and Dependence:

• Linearly Independent Vectors:
  • A set of vectors is linearly independent if none of the vectors can be expressed as a linear combination of the others.
  • In 3D, if you have three linearly independent vectors, they span the entire 3D space.
• Linearly Dependent Vectors:
  • Vectors are linearly dependent if at least one vector can be expressed as a linear combination of the others.
  • If vectors are linearly dependent, they don’t add any new dimension to the span.

Mathematical Representation:

• Linear Combination in 3D: $$\mathbf{w}=c_1\mathbf{u}+c_2\mathbf{v}+c_3\mathbf{w}$$
  • Here, $\mathbf{w}$ can represent any vector in 3D space if $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$ are linearly independent.

Puzzle:

• Basis of a Space:
  • The basis of a space is a set of linearly independent vectors that span the space.
  • Why does this definition make sense?
    • The basis provides the minimal set of vectors required to span the entire space, with each vector contributing a new dimension to the span.

This foundation sets the stage for deeper explorations into vector spaces, transformations, and matrix operations, which you’ll encounter in further studies of linear algebra.

7. Linear Transformations and Their Visualization

Key Concepts:

1. Linear Transformation:

   • A linear transformation is a special type of function in linear algebra that takes a vector as an input and outputs another vector.
   • Unlike arbitrary functions, linear transformations have two key properties:
     • Lines remain straight: No curves are introduced.
     • The origin stays fixed: The transformation does not move the origin.

2. Understanding Transformations:

   • Transformation vs. Function: Although both terms refer to input-output relationships, “transformation” suggests a geometric or visual process. For vectors, it means imagining the input vector moving to the output vector.
   • Grid Visualization: Visualizing a linear transformation involves imagining an infinite grid of points in space, with each point moving to a new location according to the transformation.

3. Visualizing Linear Transformations:

   • Grid Lines Stay Parallel: In a linear transformation, the grid lines remain straight and parallel, and the spacing between them remains consistent.
   • Effect on Space: Linear transformations can be visualized as squishing, stretching, or rotating the entire plane, but without bending lines or moving the origin.

Examples of Transformations:

1. Non-Linear Transformations:

   • If a transformation curves straight lines or moves the origin, it’s not linear. For instance, a transformation that bends a diagonal line is non-linear.

2. Linear Transformations:

   • Rotation: A common linear transformation where the entire grid rotates around the origin.
   • Scaling: Stretching or compressing the grid along certain directions.
   • Shearing: Tilting the grid while keeping lines straight and parallel.

Connection to Matrices:

• Matrix-Vector Multiplication: A key insight is that every linear transformation can be represented by a matrix. When you multiply a matrix by a vector, you’re performing a linear transformation on that vector.
  • Example:
    • Consider a 2D vector $\mathbf{v}$ and a 2x2 matrix $A$. The product $A\mathbf{v}$ gives a new vector, which is the result of applying the linear transformation represented by $A$ to $\mathbf{v}$.

Practical Visualization:

• Grid Movement: When you apply a linear transformation to all the points on a grid:
  • The points move, but the lines connecting them remain straight and evenly spaced.
  • The transformation could represent actions like rotating, stretching, or compressing the grid, but never bending it.

Summary:

• Linear transformations are central to linear algebra and are always represented by matrices. Understanding them through visualization—especially by imagining how grids are transformed—helps to intuitively grasp how matrices manipulate space. These transformations are fundamental for topics like matrix-vector multiplication, eigenvalues, and more complex vector space concepts.

Next, we’ll delve into how these transformations specifically relate to matrices and how different types of matrices (e.g., rotations, scalings) affect vectors.

8. Understanding Linear Transformations and Matrix Representation

Key Concepts:

1. Linear Transformations and Basis Vectors:

   • A linear transformation takes a vector and moves it to a new location in space.
   • To describe a linear transformation numerically, you only need to know where the basis vectors (typically $\mathbf{i}$ and $\mathbf{j}$ in 2D space) land after the transformation.

2. Tracking Vector Transformations:

   • Any vector $\mathbf{v}$ can be written as a linear combination of $\mathbf{i}$ and $\mathbf{j}$, such as: $$\mathbf{v}=x\mathbf{i}+y\mathbf{j}$$
   • After the transformation, $\mathbf{v}$ will land at a new location determined by the same linear combination of the transformed basis vectors: $$\text{New }\mathbf{v}=x(\text{New }\mathbf{i})+y(\text{New }\mathbf{j})$$

3. Example of a Transformation:

   • Suppose you have a vector $\mathbf{v}=\left(\begin{array}{c}-1\\ 2\end{array}\right)$, meaning it’s $-1$ times $\mathbf{i}$ plus $2$ times $\mathbf{j}$.
   • If the transformation moves $\mathbf{i}$ to $\left(\begin{array}{c}1\\ -2\end{array}\right)$ and $\mathbf{j}$ to $\left(\begin{array}{c}3\\ 0\end{array}\right)$, then: $$\text{New }\mathbf{v}=-1\times \left(\begin{array}{c}1\\ -2\end{array}\right)+2\times \left(\begin{array}{c}3\\ 0\end{array}\right)$$ $$\text{New }\mathbf{v}=\left(\begin{array}{c}-1\\ 2\end{array}\right)+\left(\begin{array}{c}6\\ 0\end{array}\right)=\left(\begin{array}{c}5\\ 2\end{array}\right)$$

4. Matrix Representation of Linear Transformations:

   • The linear transformation is completely described by the new positions of the basis vectors $\mathbf{i}$ and $\mathbf{j}$. This is represented by a 2x2 matrix: $$A=\left(\begin{array}{cc}1& 3\\ -2& 0\end{array}\right)$$
     • Here, the first column $\left(\begin{array}{c}1\\ -2\end{array}\right)$ shows where $\mathbf{i}$ lands, and the second column $\left(\begin{array}{c}3\\ 0\end{array}\right)$ shows where $\mathbf{j}$ lands.

5. Matrix-Vector Multiplication:

   • To find where any vector $\mathbf{v}=\left(\begin{array}{c}x\\ y\end{array}\right)$ lands after the transformation, you multiply the matrix $A$ by $\mathbf{v}$: $$A\mathbf{v}=\left(\begin{array}{cc}1& 3\\ -2& 0\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}1x+3y\\ -2x+0y\end{array}\right)=\left(\begin{array}{c}x+3y\\ -2x\end{array}\right)$$
   • This gives you the coordinates of the transformed vector.

6. Summary:

   • A 2D linear transformation can be fully described using a 2x2 matrix. Each column of the matrix corresponds to where the basis vectors $\mathbf{i}$ and $\mathbf{j}$ land after the transformation.
   • By multiplying this matrix by any vector, you can find out where the transformation will move that vector.

Understanding how linear transformations are represented by matrices and how matrix-vector multiplication works is fundamental in linear algebra. This concept is the building block for more advanced topics like eigenvalues, eigenvectors, and more complex transformations.

9. General Linear Transformations and Matrices

Key Concepts:

1. General Form of a Matrix:

   • A 2D linear transformation can be represented by a matrix with elements $a$, $b$, $c$, and $d$: $$A=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$$
   • The first column $\left(\begin{array}{c}a\\ c\end{array}\right)$ represents where the basis vector $\mathbf{i}$ lands, and the second column $\left(\begin{array}{c}b\\ d\end{array}\right)$ represents where the basis vector $\mathbf{j}$ lands after the transformation.

2. Matrix-Vector Multiplication:

   • Given a vector $\mathbf{v}=\left(\begin{array}{c}x\\ y\end{array}\right)$, the matrix transformation moves this vector to a new position: $$A\mathbf{v}=\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)=\left(\begin{array}{c}ax+by\\ cx+dy\end{array}\right)$$
   • This multiplication represents taking a linear combination of the transformed basis vectors, scaled by the components of $\mathbf{v}$.

3. Interpreting the Matrix as a Transformation:

   • Instead of memorizing the formula, think of the matrix columns as where the basis vectors $\mathbf{i}$ and $\mathbf{j}$ end up after the transformation.
   • For example:
     • If $A=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$, this matrix represents a 90-degree counterclockwise rotation because $\mathbf{i}$ lands at $\left(\begin{array}{c}0\\ 1\end{array}\right)$ and $\mathbf{j}$ lands at $\left(\begin{array}{c}-1\\ 0\end{array}\right)$.

4. Examples of Specific Transformations:

   • Rotation: Rotating space by 90 degrees counterclockwise gives the matrix: $$A=\left(\begin{array}{cc}0& -1\\ 1& 0\end{array}\right)$$
     • This matrix rotates any vector in 2D space by 90 degrees counterclockwise.
   • Shear Transformation: A shear transformation might keep $\mathbf{i}$ fixed and move $\mathbf{j}$ to a new position: $$A=\left(\begin{array}{cc}1& 1\\ 0& 1\end{array}\right)$$
     • Here, $\mathbf{i}$ remains at $\left(\begin{array}{c}1\\ 0\end{array}\right)$ while $\mathbf{j}$ moves to $\left(\begin{array}{c}1\\ 1\end{array}\right)$, resulting in a skewing effect.

5. Reverse Engineering Transformations:

   • Given a matrix $A=\left(\begin{array}{cc}1& 3\\ 2& 1\end{array}\right)$, you can visualize the transformation by imagining how $\mathbf{i}$ moves to $\left(\begin{array}{c}1\\ 2\end{array}\right)$ and $\mathbf{j}$ moves to $\left(\begin{array}{c}3\\ 1\end{array}\right)$.
   • If the columns of the matrix are linearly dependent (e.g., one column is a multiple of the other), the transformation squishes space onto a line.

6. Summary:

   • Linear transformations keep grid lines parallel, evenly spaced, and the origin fixed.
   • The matrix representing a transformation contains the coordinates of where the basis vectors $\mathbf{i}$ and $\mathbf{j}$ land.
   • Matrix-vector multiplication then tells you where any vector will land after the transformation.

Important Takeaways:

• Matrices are not just arrays of numbers; they represent transformations of space.
• Understanding matrix transformations deeply enhances your grasp of linear algebra, giving you tools to visualize and manipulate vectors and spaces.

By internalizing these concepts, you’ll be well-equipped to tackle more complex ideas in linear algebra, such as eigenvalues, eigenvectors, and higher-dimensional transformations.