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Linear Transformations and Matrices
Learn how linear transformations move vectors, how basis vectors determine the transformation, and why matrix-vector multiplication computes its result.
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What the lecture covers
A transformation maps input vectors to output vectors. Visualizing vectors as points moving through space makes it easier to see how a transformation acts overall. In two dimensions, linear transformations keep the origin fixed and send straight lines to straight lines; equivalently, they preserve the parallel, evenly spaced structure of a grid.
A linear transformation is fully determined by where it sends the two basis vectors. Any vector is a linear combination of those basis vectors, and its image is the same combination of their images. A matrix records those images as its columns, so multiplying a matrix by a vector calculates the corresponding combination. Examples include a 90-degree rotation and a shear. If the basis vectors’ images are linearly dependent, the transformation compresses the plane onto a line. Thinking of matrices as transformations of space gives matrix-vector multiplication a geometric meaning and helps make later linear algebra topics easier to understand.
Key ideas
A transformation maps input vectors to output vectors, and it can be visualized as points moving through space.
A linear transformation fixes the origin and keeps lines straight, preserving the grid’s parallel and evenly spaced structure.
In two dimensions, the images of the two basis vectors determine where every vector goes.
A vector’s image is found by taking the same linear combination of the images of the basis vectors.
A 2×2 matrix records the coordinates of the two basis vectors’ images in its columns.
Matrix-vector multiplication computes the linear combination of the matrix columns specified by the vector’s coordinates.
Rotations and shears can be represented by matrices whose columns show where the basis vectors land.
If the images of the basis vectors are linearly dependent, the transformation compresses the plane onto a line.
Sample questions
A transformation of the plane keeps every straight line straight but moves the origin to a different point. Which conclusion follows from the visual criteria for linear transformations?
AIt is linear only if the origin moves along a straight line.
BIt is not linear, because a linear transformation must leave the origin fixed.
CIt is linear, because keeping straight lines straight is sufficient.
DIt is not linear, because linear transformations must keep every point fixed.
Show answer
Correct answer: B. A linear transformation must both preserve straightness of lines and keep the origin fixed. Since the origin moves, the transformation fails a required condition.
For a linear transformation of the plane, why does knowing the images of the two coordinate basis vectors determine the image of every vector?
AThe transformation acts only on vectors that lie along the coordinate axes.
BThe transformation must send every vector to one of the two basis-vector images.
CThe lengths and angles of all vectors are fixed once the basis-vector images are known.
DEvery vector is a linear combination of the basis vectors, and the transformation preserves those combinations.
Show answer
Correct answer: D. Any vector can be expressed using the basis vectors, and linearity carries its coefficients through to the corresponding combination of their images.