Grade 12

Advanced

8 min

Lesson 10 of 12

Vectors and Linear Transformations

Not Started

Explore how matrices act as functions that transform vectors in 2D and 3D space.

Ever wonder how a computer animator turns a 2D sketch into a spinning 3D superhero? The secret isn't magic—it's a mathematical 'machine' called a linear transformation.

Learning Objectives

1.Represent geometric transformations like rotation and scaling as matrix operations
2.Calculate the image of a vector under a specific linear transformation
3.Identify eigenvectors and eigenvalues for simple 2x2 transformation matrices

Matrices as Functional Machines

In linear algebra, we don't just view matrices as static boxes of numbers. Instead, we treat a matrix $A$ as a function that takes an input vector $\vec{v}$ and transforms it into an output vector $\vec{w}$ . This is written as $T(\vec{v}) = A\vec{v}$ . For a transformation to be linear, it must satisfy two rules: it must keep the origin fixed at $(0,0)$ , and it must keep all grid lines parallel and evenly spaced. Whether you are zooming in on a map (scaling) or turning a steering wheel (rotation), you are applying a linear transformation to the underlying coordinate space.

Let's apply a scaling transformation to a vector. 1. Suppose we have a matrix

A = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix}

and a vector

\vec{v} = \begin{pmatrix} 1 \\ 4 \end{pmatrix}

. 2. To find the image, multiply the matrix by the vector:

A\vec{v} = \begin{pmatrix} 3 & 0 \\ 0 & 2 \end{pmatrix} \begin{pmatrix} 1 \\ 4 \end{pmatrix} = \begin{pmatrix} (3 \times 1) + (0 \times 4) \\ (0 \times 1) + (2 \times 4) \end{pmatrix} = \begin{pmatrix} 3 \\ 8 \end{pmatrix}

3. The vector was stretched by a factor of 3 horizontally and 2 vertically.

Quick Check

If a matrix $A$ transforms vector $\vec{v}$ into vector $\vec{w}$ , what is the formal term for $\vec{w}$ ?

Think about what a mirror produces.

Answer

The image of vector $\vec{v}$ under the transformation $A$ .

The Power of Rotation

To rotate a vector by an angle

\theta

counter-clockwise, we use a specific Rotation Matrix. This matrix is derived by tracking where the standard basis vectors

\hat{i} = (1,0)

and

\hat{j} = (0,1)

land after the turn. The general form is:

R_{\theta} = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}

By plugging in any angle, you can compute exactly where any point in space will land. This is the fundamental math behind every 3D rotation in modern robotics and video game engines.

Rotate the vector $\vec{v} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$ by $90^\circ$ counter-clockwise. 1. Set $\theta = 90^\circ$ . Note that $\cos(90^\circ) = 0$ and $\sin(90^\circ) = 1$ . 2. Construct the matrix: $R_{90} = \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}$ . 3. Multiply: $\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$ . 4. The vector moved from the x-axis to the y-axis, a perfect $90^\circ$ rotation.

Quick Check

What is the determinant of a standard rotation matrix?

Calculate

\cos^2\theta - (-\sin^2\theta)

Answer

Eigenvectors: The Invariant Lines

Most vectors change direction when transformed. However, for every transformation, there are special vectors that stay on their original span—they only get longer or shorter. These are Eigenvectors. The factor by which they are stretched is the Eigenvalue ( $\lambda$ ). The relationship is defined by the equation: $A\vec{v} = \lambda\vec{v}$ . To find these, we solve the characteristic equation: $\det(A - \lambda I) = 0$ . Finding these 'fixed' directions is crucial for understanding complex systems, from bridge vibrations to Google's search algorithm.

Find the eigenvalues for

A = \begin{pmatrix} 4 & 1 \\ 2 & 3 \end{pmatrix}

. 1. Set up

\det(A - \lambda I) = 0

\det\begin{pmatrix} 4-\lambda & 1 \\ 2 & 3-\lambda \end{pmatrix} = 0

2. Expand the determinant:

(4-\lambda)(3-\lambda) - (1 \times 2) = 0

. 3. Simplify to a quadratic:

\lambda^2 - 7\lambda + 10 = 0

. 4. Factor:

(\lambda - 5)(\lambda - 2) = 0

. 5. The eigenvalues are

\lambda_1 = 5

and

\lambda_2 = 2

Key Takeaways

1Matrices act as functions that map input vectors to output images.
2The rotation matrix $R_{\theta}$ uses trigonometric values to move vectors around the origin.
3Eigenvectors are vectors whose direction remains unchanged during a transformation.
4Eigenvalues represent the scaling factor of their corresponding eigenvectors.

Quick Check

Multiple Choice

Which matrix represents a transformation that doubles the size of every vector in both dimensions?

Multiple Choice

If $A\vec{v} = 3\vec{v}$ , what is the eigenvalue associated with the eigenvector $\vec{v}$ ?

True/False

A linear transformation can curve straight lines as long as the origin remains fixed.

What's Next?

Review Tomorrow

In 24 hours, try to write down the 2x2 rotation matrix from memory and explain what an eigenvector is to a friend.

Practice Activity

Try this on your own: Calculate the image of $\vec{v} = \begin{pmatrix} 2 \\ 3 \end{pmatrix}$ using a $180^\circ$ rotation matrix.

Browse

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Vectors and Linear Transformations

Learning Objectives

Matrices as Functional Machines

The Power of Rotation

Eigenvectors: The Invariant Lines

Key Takeaways

Quick Check

What's Next?

Vectors and Linear Transformations

Learning Objectives

Matrices as Functional Machines

The Power of Rotation

Eigenvectors: The Invariant Lines

Key Takeaways

Quick Check

What's Next?