30. Linear Transformations and Their Matrices
Watch on YouTube →
Overview
Linear transformations act on vectors while preserving addition and scalar multiplication, and choosing input and output bases turns each transformation into a matrix. The lecture shows how the matrix columns encode the transformed basis vectors, using projections, rotations, a map from R^3 to R^2, and differentiation of quadratic polynomials.
Key takeaways
- A linear transformation is determined completely by its action on an input basis because linearity fixes its output on every combination of basis vectors.
- A transformation from R^n to R^m is represented by an m-by-n matrix once input and output bases are specified.
- Each matrix column is the output of the corresponding input basis vector, expressed in the output basis.
- The same projection has matrix diag(1, 0) in a basis aligned with the projected line and its perpendicular, but matrix [[1/2, 1/2], [1/2, 1/2]] in the standard basis for projection onto the 45-degree line.
- Differentiation on polynomials of degree at most two has matrix [[0, 1, 0], [0, 0, 2]] when the input basis is (1, x, x^2) and the output basis is (1, x).
- A nonzero translation fails linearity because it does not map zero to zero, while the vector-length map fails scalar compatibility for negative scalars.
Chapters
- A transformation maps each input vector to an output vector, and can be described without coordinates or a matrix.
- Linearity requires T(v + w) = T(v) + T(w) and T(cv) = cT(v) for vectors v, w and scalar c.
- A projection onto a line is linear because scaling or adding inputs scales or adds their projections.
- The shift T(v) = v + v0 is not linear: doubling v does not double the fixed offset v0.
- Every linear transformation must map the zero vector to zero, a condition violated by a nonzero shift.
- The length map from R^3 to R is not linear because the length of -2v is 2 times the length of v, not -2 times it.
- A 45-degree rotation is linear: rotating a scaled or summed vector gives the same result as scaling or summing the rotated vectors.
- Applying a linear transformation to every point of a house outline shows how the whole shape changes, not just one vector.
- The matrix diag(1, -1) preserves each x-coordinate and reverses each y-coordinate, flipping a house upside down.
- For T(v) = Av to map inputs in R^3 to outputs in R^2, A must be a 2-by-3 matrix.
- Every matrix of the appropriate size defines a linear transformation because matrix multiplication preserves vector sums and scalar multiples.
- The goal is to show that every linear transformation can be represented by a matrix once bases are chosen.
- Knowing T on one vector gives limited information, while knowing T on independent basis vectors determines its action on their entire span.
- For an input basis v1 through vn, every vector v = c1v1 + ... + cnvn has output T(v) = c1T(v1) + ... + cnT(vn).
- Linearity means the outputs on the basis vectors fully determine the transformation on the input space.
- Coordinates are the coefficients expressing a vector as a combination of basis vectors, so changing the basis changes the coordinates.
- A transformation from R^n to R^m requires an input basis and an output basis to represent inputs and outputs numerically.
- The representing matrix maps the input coordinates to the output coordinates.
- For projection onto a line, choose one basis vector along the line and another perpendicular to it; projection keeps the first and sends the second to zero.
- In this eigenvector basis, the projection matrix is diagonal, with eigenvalues 1 and 0.
- For projection onto the 45-degree line in the standard basis, the matrix is [[1/2, 1/2], [1/2, 1/2]], which is symmetric and satisfies P^2 = P.
- To construct column j of a transformation matrix, apply T to the jth input basis vector.
- Express that output as a combination of the output basis vectors; its coefficients become column j.
- Multiplying the matrix by input coordinates combines its columns and produces the output coordinates for any input.
- Differentiation maps the three-dimensional polynomial space spanned by 1, x, and x^2 into the two-dimensional space spanned by 1 and x.
- For p(x) = c1 + c2x + c3x^2, the derivative is c2 + 2c3x.
- With the stated input and output bases, the derivative matrix is [[0, 1, 0], [0, 0, 2]].
- Matrix products represent compositions of linear transformations, and an inverse matrix represents an inverse transformation when one exists.
Transcript excerpt
0:07 OK, this is the lecture on linear transformations. Actually, linear algebra courses used to begin with this lecture, so you could say I'm beginning this course again by talking about linear transformations. In a lot of courses, those come first before matrices. The idea of a linear transformation makes sense without a matrix, and physicists and other -- some people like it better that way. They don't like coordinates. They don't want those numbers. They want to see what's going on with the whole space. But, for most of us, in the end, if we're
0:59 going to compute anything, we introduce coordinates, and then every linear transformation will lead us to a matrix. And then, to all the things that we've done about null space and row space, and determinant, and eigenvalues -- all will come from the matrix. But, behind it -- in other words, behind this is the idea of a linear transformation. Let me give an example of a linear transformation. So, example. Example one. A projection. I can describe a projection without telling you any matrix,
1:53 anything about any matrix. I can describe a projection, say, this will be a linear transformation that takes, say, all of R^2, every vector in the plane, into a vector in the plane. And this is the way people describe, a mapping. It takes every vector, and so, by what rule? So, what's the rule, is, I take a -- so here's the plane, this is going to be my line, my line through my line, and I'm going to project every vector onto that line. So if I take a vector like b -- or let me call the vector v for the moment --
2:43 the projection -- the linear transformation is going to produce this vector as T(v).
This is the start of the transcript. Save this video free for the whole transcript, searchable and time-coded.
Related notes
Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, MIT OpenCourseWare.