Home › Lectures › 3Blue1Brown — Essence of linear algebra
Nonsquare Matrices as Transformations Between Dimensions
The lecture explains how rectangular matrices represent linear transformations between spaces of different dimensions and how to read their sizes geometrically.
Free, sign in with Google. A ready quiz does not use your hourly limit.
What the lecture covers
A linear transformation can map vectors between spaces of different dimensions, such as from 2D to 3D. It still preserves the origin and keeps grid lines parallel and evenly spaced. To represent the transformation with a matrix, place the coordinates of each input basis vector’s image in a column. Thus, a transformation from 2D to 3D has a 3-by-2 matrix: two columns for the input basis vectors and three rows for the coordinates of their outputs. In the example, the outputs span a plane through the origin in 3D, so the matrix has full rank.
The matrix dimensions reveal the direction of a transformation: columns count input dimensions, while rows count output dimensions. A 2-by-3 matrix maps 3D inputs to 2D outputs. A 1-by-2 matrix maps 2D vectors to numbers on the number line; in this case, linearity can be understood as preserving equal spacing among points. Such transformations are connected to the dot product, which the next lecture will explore.
Key ideas
Linear transformations can map vectors between spaces of different dimensions while preserving the origin and the structure of evenly spaced, parallel grid lines.
The columns of a transformation matrix are the coordinates of the images of the input basis vectors.
A 3-by-2 matrix represents a map from 2D to 3D, and its example’s outputs span a plane through the origin.
For a rectangular matrix, the number of columns gives the input dimension and the number of rows gives the output dimension.
A 2-by-3 matrix maps vectors from 3D space to 2D space.
A 1-by-2 matrix maps 2D vectors to the number line, where linearity can be seen as preserving equal spacing.
Transformations from 2D to 1D have a close connection to the dot product.
Sample questions
A linear transformation maps input vectors from a two-dimensional vector space to output vectors in a separate three-dimensional vector space. Which interpretation best respects the distinction between the two spaces?
AInputs and outputs must be points in one shared space because a transformation connects them.
BEach input and its output are vectors in their own spaces; the transformation relates them without making them points in one shared space.
CThe output is the same input vector after an extra coordinate is added, so both belong to the input space.
DThe output space is merely a visual copy of the input space, so the two vectors have the same type.
Show answer
Correct answer: B. A transformation pairs vectors from its domain with vectors in its codomain; relating them does not make them members of the same space.
A linear transformation sends the first standard basis vector to (2, -1) and the second to (3, 4). Which matrix represents the transformation when vectors are written as columns?
A[[3, 2], [4, -1]]
B[[2, 3], [-1, 4]]
C[[-1, 4], [2, 3]]
D[[2, -1], [3, 4]]
Show answer
Correct answer: B. Each image becomes a column in the same order as its corresponding basis vector.