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What the lecture covers
A three-dimensional linear transformation moves points in space while keeping grid lines parallel and evenly spaced and leaving the origin fixed. As in two dimensions, the transformation is fully determined by where the basis vectors land. In 3D, these are the unit vectors along the x, y, and z axes. Their output coordinates become the columns of a 3×3 matrix, so nine numbers specify the transformation.
For example, a 90-degree rotation around the y-axis sends the x-basis vector to the negative z direction, leaves the y-basis vector unchanged, and sends the z-basis vector to the x direction. To transform a vector with coordinates (x, y, z), scale each matrix column by the corresponding coordinate and add the results. Matrix multiplication represents successive transformations: apply the transformation encoded by the right-hand matrix first, then the one encoded by the left. This makes it possible to describe complex 3D rotations as combinations of simpler ones, a useful approach in areas such as computer graphics and robotics.
Key ideas
A 3D linear transformation moves space while preserving parallel, evenly spaced grid lines and fixing the origin.
A 3D transformation is determined by the images of the three standard basis vectors.
The coordinates of the transformed basis vectors form the columns of a 3×3 matrix.
A 90-degree rotation about the y-axis maps the x basis vector to negative z, keeps y fixed, and maps z to x.
To transform a vector, multiply each matrix column by the corresponding input coordinate and add the results.
Multiplying two matrices means applying the right-hand transformation first and the left-hand transformation second.
Composing simpler rotations helps describe 3D rotations and is useful in computer graphics and robotics.
Sample questions
In three-dimensional space, a point P can be represented by the position vector whose tail is at the origin O and whose tip is at P. If a transformation fixes O, how does this representation describe the image of P?
AThe transformed position vector has its tip at the image of P.
BThe point P stays fixed because its position vector begins at the fixed origin.
CThe transformed position vector has its tail at the image of P.
DThe image of P is determined only by the distance from P to the origin.
Show answer
Correct answer: A. With the origin fixed, the point is represented by the tip of its position vector, so moving the vector’s tip gives the point’s image.
For a linear transformation of three-dimensional space, suppose two transformations send each of the three standard basis vectors to the same output. What follows for their outputs on any vector in three-dimensional space?
AThey may differ on vectors that are not basis vectors.
BThey produce the same output for every vector.
CThey produce the same output only for vectors of unit length.
DThey produce the same output only when all three basis vectors remain unchanged.
Show answer
Correct answer: B. Every vector in three-dimensional space is a linear combination of the three standard basis vectors, and linearity means matching outputs on those basis vectors determines the output for every such combination.