Chapter 5: Three-dimensional linear transformations | Essence of Linear Algebra
BeginnerKey Summary
- •Three-dimensional linear transformations change the whole 3D space while keeping the origin fixed and all grid lines straight and parallel. This is just like in 2D, but now there are three axes: x, y, and z. These transformations stretch, rotate, shear (slant), or reflect the space without bending or curving it.
- •Any 3D linear transformation is fully described by what it does to the three basis vectors: i-hat (1,0,0), j-hat (0,1,0), and k-hat (0,0,1). Once you know where these three arrows move, you know what happens to every other vector. This is because every vector is a mix (linear combination) of i-hat, j-hat, and k-hat.
- •You can write a 3D linear transformation as a 3x3 matrix. The first column is the new position of i-hat, the second column is the new position of j-hat, and the third column is the new position of k-hat. Multiplying this matrix by any vector gives the transformed vector.
- •Matrix–vector multiplication is a fast way to compute the exact linear combination of the transformed basis vectors. If a vector v equals a*i-hat + b*j-hat + c*k-hat, then T(v) equals a*T(i-hat) + b*T(j-hat) + c*T(k-hat). This is how linearity works in 3D.
- •Example: Suppose T(i-hat) = (1, -1, 1), T(j-hat) = (-1, -1, 1), and T(k-hat) = (1, 1, 1). For v = 3*i-hat - 2*j-hat + 5*k-hat, the transformed result is 10, 0, 6. This comes from adding 3*(1,-1,1) + (-2)*(-1,-1,1) + 5*(1,1,1).
- •Matrix multiplication matches doing one linear transformation after another. If you first rotate and then shear, multiply the shear matrix by the rotation matrix in that order to get a single matrix for the combined effect. The order matters because the output of the first becomes the input of the second.
- •
Why This Lecture Matters
Understanding 3D linear transformations is essential for anyone who needs to control or predict motion and shape in space. Software engineers in graphics use these matrices to rotate, scale, and position models and cameras smoothly. Roboticists depend on them to move arms and tools through precise paths, stacking many small transformations with correct order. Physicists use them to switch coordinate systems and analyze rotations and symmetries without changing the underlying physics. This knowledge solves practical problems like how to rotate an object around an axis, how to combine multiple moves into one efficient step, and how to avoid mistakes from the wrong order of operations. It also helps you interpret matrices quickly: columns show new axes, determinants reveal volume changes, and orthogonality signals pure rotation. In real projects, this saves time and reduces bugs because you can design, check, and debug transformations using clear geometric meaning instead of guesswork. For career growth, mastering these ideas opens doors in game development, animation, AR/VR, robotics control, simulation, and scientific computing. These fields all rely on composing transformations and using rotation matrices daily. Industry values people who can move easily between the algebra (matrices) and the geometry (space motion). With this skill, you can build features faster, communicate clearly with teammates, and deliver stable, correct, and visually accurate results.
Lecture Summary
Tap terms for definitions01Overview
This lesson teaches how three-dimensional linear transformations work and how they closely mirror, yet extend beyond, the familiar ideas from two-dimensional space. A linear transformation is a rule that changes every point in space, while keeping the origin fixed and straight lines straight. In 3D, this rule acts on the entire grid of space: it can stretch, squish, rotate, shear (slant), or reflect, but it never bends curves or moves the origin. The key idea is that the whole transformation is fully determined by what it does to the three standard basis vectors: i-hat (1,0,0), j-hat (0,1,0), and k-hat (0,0,1). Because any vector can be written as a combination of these three, knowing where they go tells you where everything else goes.
You will learn to represent these transformations with 3x3 matrices. Each column of the matrix is simply the new position of one basis vector: the first column is the image of i-hat, the second is of j-hat, and the third is of k-hat. Multiplying this matrix by a vector gives the transformed vector. This multiplication is not magic—it's just a compact way of adding together scaled versions of the transformed basis vectors according to the vector's coordinates. The lesson includes a concrete example: given T(i-hat) = (1, -1, 1), T(j-hat) = (-1, -1, 1), and T(k-hat) = (1, 1, 1), the vector v = 3i - 2j + 5k transforms to (10, 0, 6) by linearity.
A major theme is composing transformations. Doing one linear transformation after another is the same as multiplying their matrices, in the correct order. For instance, if you first rotate around an axis and then apply a shear, you multiply the shear's matrix by the rotation's matrix to get the combined effect. Importantly, the order matters—especially with 3D rotations. Rotating around the x-axis and then the y-axis is usually not the same result as doing them in the opposite order. This shows that matrix multiplication is not commutative (you can't swap the order without changing the result).
Rotations about the coordinate axes are a special and very useful family of transformations. The rotation around the z-axis by an angle θ keeps the k-hat vector unchanged and spins i-hat and j-hat within the xy-plane. Its matrix has a neat form: the first column is (cosθ, sinθ, 0), the second column is (−sinθ, cosθ, 0), and the third column is (0, 0, 1). Similar patterns describe rotations about the x-axis and y-axis, each leaving its own axis' basis vector fixed. Understanding these rotation matrices is core to many applications, from moving 3D models in graphics to controlling robot joints.
This content is designed for learners who know the basics of vectors and 2D linear transformations—like what a matrix is, what a vector is, and how matrix–vector multiplication works. You don't need advanced math, but being comfortable with coordinates and a bit of trigonometry (sine and cosine) will help when dealing with rotation angles. The lesson fits beginners stepping into 3D from 2D, as well as anyone needing a solid, visual way to think about 3D motions that preserve straight lines and keep the origin fixed.
After finishing, you'll be able to build and read 3x3 matrices that represent 3D linear transformations. You'll compute how any vector moves under a given transformation using matrix–vector multiplication. You'll create rotation matrices about the x, y, and z axes for any angle, and you'll combine multiple transformations by multiplying their matrices in the right order. Most of all, you'll build intuition: watching how the basis vectors move makes it clear how the entire grid of space moves.
The lesson flows from a reminder of 2D linear transformations to their 3D counterparts, highlighting what stays the same and what changes. It then shows how knowing the transformed basis vectors determines the whole transformation, illustrates this with a worked numeric example, and explains how to build the 3x3 matrix from the columns. Next, it covers the composition of transformations via matrix multiplication, and demonstrates why order matters in 3D, especially with rotations. Finally, it focuses on rotation around an axis, constructing the z-axis rotation matrix and noting similar forms for x- and y-axis rotations. It closes by encouraging hands-on exploration—change matrix entries, visualize how the grid moves, and connect these ideas to practical fields like graphics, robotics, and physics.
Key Takeaways
- ✓Always build a 3D transformation matrix from the images of i, j, and k. Put T(i), T(j), and T(k) as the first, second, and third columns. This guarantees A·i = T(i), A·j = T(j), and A·k = T(k). It removes guessing and makes the transformation unambiguous.
- ✓Compute A·v by mixing columns with v’s coordinates. Think of v = (a,b,c) as weights on the columns: a·col1 + b·col2 + c·col3. This method is fast and matches the geometric idea of linear combinations. Use it to transform any vector reliably.
- ✓Plan composition carefully because order matters. To do A then B, multiply to get B·A. Remember the rightmost matrix acts first. Test simple inputs to confirm your ordering is correct.
- ✓Use standard rotation matrices around axes for quick setups. Rz(θ), Rx(θ), and Ry(θ) follow simple cosine–sine patterns and keep their rotation axis fixed. They preserve lengths and angles, making them safe building blocks. Combine them for any desired orientation.
- ✓Watch the basis vectors to understand the whole transform. If you can predict where i, j, and k go, you can predict where everything goes. Visualize these arrows and a unit cube to grasp shear and rotation effects. This speeds up debugging and design.
- ✓Check angle units: many functions expect radians. Convert degrees with θrad = θdeg·π/180. A wrong unit silently creates wrong rotations. Make this a habit whenever you use trig functions.
- ✓Use determinants to sanity-check your matrix. |det(A)| tells how volume changes, and det(A) = 0 warns of flattening and non-invertibility. Negative det means a reflection is involved. These quick checks catch major errors early.
Glossary
Linear transformation
A rule that changes every vector while keeping the origin fixed and straight lines straight. It follows two laws: add-then-transform equals transform-then-add, and scale-then-transform equals transform-then-scale. In symbols, T(a·u + b·v) = a·T(u) + b·T(v). This makes the space bend in a uniform, predictable way. It never curves lines or moves (0,0,0).
Basis vectors (i-hat, j-hat, k-hat)
The three unit arrows along the x, y, and z axes: i = (1,0,0), j = (0,1,0), k = (0,0,1). Any vector can be written by mixing these three with numbers. They are the building blocks of 3D coordinates. Knowing where a transformation sends these tells you everything about the transformation.
Linear combination
A weighted mix of vectors using addition and scaling. For vectors u, v and numbers a, b, the linear combination is a·u + b·v. Any 3D vector is a combination of i, j, and k. This idea lets you build all vectors from a small set.
Matrix
A grid of numbers that can represent a linear transformation. In 3D, it has 3 rows and 3 columns for a linear map from R^3 to R^3. Each column can be the image of a basis vector. Multiplying a matrix by a vector applies the transformation.