Unit 9: Linear Transformations and Change of Basis
A matrix as a machine that moves the whole plane at once, and how the same arrow looks different described in two different grids.
9 topics · 33 questions
Each topic is a path, not a menu: work down it in order and every question builds on the one above. Click any of them to watch the answer draw itself.
What a Linear Transformation Is
- How it worksHow do preserving addition and preserving scaling together define exactly what makes a transformation linear?
- Common mistakeWhy isn't shifting every point over by a fixed amount a linear transformation?
- Going furtherWhat happens to the origin under any linear transformation at all, no matter which one it is?
Your turn1 of 2
The map (x, y) to (x + 7, y) shifts every point sideways by k. Why isn't this a linear transformation?The Matrix of a Linear Transformation
- How it worksHow do the columns of a transformation's matrix come from where it sends each basis vector?
- Worked exampleFind the matrix of the linear transformation that sends (1, 0) to (2, 1) and (0, 1) to (0, 3).
- Worked exampleFind the 2 by 2 matrix that carries (1, 0) to (3, -1) and (0, 1) to (1, 2).
- Common mistakeWhy doesn't every transformation between vector spaces have to be linear?
Your turn1 of 2
A linear transformation sends (1, 0) to (3, 2) and (0, 1) to (7, -4). Find the first entry of where it sends the vector (-7, -5).Rotation, Reflection, and Scaling as Matrices
- How it worksHow do the cosine and sine of an angle build the matrix that rotates every vector in the plane?
- Worked exampleFind the matrix that rotates every vector in the plane by 90 degrees counterclockwise.
- Worked exampleFind the matrix that reflects every vector in the plane across the x-axis.
- Going furtherWhat happens when you apply a rotation matrix twice in a row?
Your turn1 of 2
A vector (4, 3) is rotated counterclockwise by an angle of 5.14 radians. Find the x-coordinate of the result.Shear Transformations
- How it worksHow does a shear matrix move each point sideways by an amount that depends on its height?
- Worked exampleFind the matrix that shears the plane by adding twice the y-coordinate onto the x-coordinate.
- Common mistakeWhy doesn't a shear transformation change the area of a shape it's applied to?
- Going furtherWhat happens when the same shear transformation is applied twice in a row?
Your turn1 of 2
A shear transformation adds 2 times the y-coordinate onto the x-coordinate, leaving y unchanged. Find the new x-coordinate of the point (3, 8).Linear Transformations Between Different Dimensions
- How it worksHow does a non-square matrix carry a linear transformation between spaces of two different dimensions?
- Worked exampleFind the matrix of the linear transformation from 3-space to the plane that keeps only the first two coordinates.
- Going furtherWhat happens to the kernel of a transformation that drops a vector from 3-space down to the plane?
Your turn1 of 2
A transformation from 3-space to the plane keeps only the first two coordinates. Find the sum of the two output coordinates of the vector (4, 3, 8).Coordinates Relative to a Basis
- How it worksHow does writing a vector as a combination of basis vectors produce its coordinates in that basis?
- Worked exampleFind the coordinates of the vector (5, 4) relative to the basis (1, 1) and (1, -1).
- Common mistakeWhy doesn't a vector's coordinates changing in a new basis mean the vector itself changed?
Your turn1 of 2
Find the first coordinate of the vector (4, 3) relative to the basis (1, 1) and (1, -1).Change-of-Basis Matrices
- How it worksHow does the change-of-basis matrix built from the new basis vectors convert coordinates between bases?
- Worked exampleFind the change-of-basis matrix converting coordinates in the basis (1, 1), (1, -1) back into standard coordinates.
- Going furtherWhat happens to a linear transformation's matrix once you change to a basis made of its own eigenvectors?
- Ties togetherHow does describing a transformation in the basis of its own eigenvectors turn its matrix into a simple diagonal one?
Your turn1 of 2
The change-of-basis matrix converting coordinates in the basis (3, 2), (7, -4) back into standard coordinates has those vectors as its columns. Find the first standard coordinate of the vector whose coordinates in this basis are (-7, -5).Composing Linear Transformations
- How it worksHow does multiplying the two transformations' matrices together produce the matrix for doing both in sequence?
- Worked exampleFind the matrix for rotating by 90 degrees and then reflecting across the x-axis, in that order.
- Common mistakeWhy does the order you compose two transformations in usually matter?
- Going furtherWhat happens to the composed matrix when the same linear transformation is applied three times in a row?
Your turn1 of 2
Find the (1,1) entry of the matrix for applying the transformation with matrix [[3, 2], [7, -4]] first, then the transformation with matrix [[-7, -5], [3, -8]].The Kernel and Range of a Linear Transformation
- How it worksHow does the kernel collect everything a transformation crushes to zero, while the range collects everything it can produce?
- Worked exampleFind the kernel and range of the linear transformation given by the matrix [[1, 1], [2, 2]].
- Common mistakeWhy doesn't a transformation with a large kernel also have to have a large range?
- Ties togetherHow does the kernel and range of a transformation match up exactly with the null space and column space of its matrix?
Your turn1 of 2
A linear transformation is given by a matrix whose second column is exactly 2 times its first column, (6, 7). Find the x-coordinate of a nonzero kernel vector whose y-coordinate is 1.