https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:matrices-as-transformations/v/matrices-as-transformations-of-the-plane 2022-10-07T00:22:18.752150461Z Matrices as transformations of the plane We can think of a 2X2 matrix as describing a special kind of transformation of the plane (called "linear transformation"). By telling us where the vectors [1,0] and [0,1] are mapped to, we can figure out where any other vector is mapped to. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/944b2fbb-8837-4220-8b6e-19d201144c76_1280_720_base.png Matrices as transformations of the plane We can think of a 2X2 matrix as describing a special kind of transformation of the plane (called "linear transformation"). By telling us where the vectors [1,0] and [0,1] are mapped to, we can figure out where any other vector is mapped to. https://cdn.kastatic.org/ka-youtube-converted/PtjdhADnJi4.mp4/PtjdhADnJi4.mp4 492 Matrices as transformations of the plane https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:matrices-as-transformations/v/working-with-matrices-as-transformations-of-the-plane 2022-10-07T00:22:18.752150461Z Working with matrices as transformations of the plane Once we know how 2X2 matrices define transformations of the plane, we can connect geometric transformations like rotations, reflections, and dilations to specific matrices. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/3db36e2f-f2f2-4717-966d-9f714e9b7428_1280_720_base.png Working with matrices as transformations of the plane Once we know how 2X2 matrices define transformations of the plane, we can connect geometric transformations like rotations, reflections, and dilations to specific matrices. https://cdn.kastatic.org/ka-youtube-converted/vCkvVAvmJwY.mp4/vCkvVAvmJwY.mp4 255 Matrices as transformations of the plane https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:matrices-as-transformations/v/intro-to-determinants 2022-10-07T00:22:18.752150461Z Intro to determinant notation and computation Matrix determinants are easy to define and hard to understand. So let's start with defining them and introducing related notation. In other videos we will learn what they mean and how to use them. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/f2347269-718e-4218-b496-7eb72e71edf5_1280_720_base.png Intro to determinant notation and computation Matrix determinants are easy to define and hard to understand. So let's start with defining them and introducing related notation. In other videos we will learn what they mean and how to use them. https://cdn.kastatic.org/ka-youtube-converted/ZAkmBOCiNqU.mp4/ZAkmBOCiNqU.mp4 206 Matrices as transformations of the plane https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:matrices-as-transformations/v/interpreting-determinants-in-terms-of-area 2022-10-07T00:22:18.752150461Z Interpreting determinants in terms of area The determinant of a 2X2 matrix tells us what the area of the image of a unit square would be under the matrix transformation. This, in turn, allows us to tell what the area of the image of any figure would be under the transformation. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/ad5cbf15-6db2-4532-a096-7afbf0b8d21f_1280_720_base.png Interpreting determinants in terms of area The determinant of a 2X2 matrix tells us what the area of the image of a unit square would be under the matrix transformation. This, in turn, allows us to tell what the area of the image of any figure would be under the transformation. https://cdn.kastatic.org/ka-youtube-converted/ER_96Mx-KEE.mp4/ER_96Mx-KEE.mp4 259 Matrices as transformations of the plane https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:matrices-as-transformations/v/finding-area-of-figure-after-transformation-using-determinant 2022-10-07T00:22:18.752150461Z Finding area of figure after transformation using determinant In this worked example, Sal finds the area of the image of a rectangle after a transformation defined by a given matrix. This is done by finding the area of the pre-image and multiplying by the matrix's determinant. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/86976ed8-8ee5-41c8-9244-7003ab2330cc_1280_720_base.png Finding area of figure after transformation using determinant In this worked example, Sal finds the area of the image of a rectangle after a transformation defined by a given matrix. This is done by finding the area of the pre-image and multiplying by the matrix's determinant. https://cdn.kastatic.org/ka-youtube-converted/aVuCoUxt3vc.mp4/aVuCoUxt3vc.mp4 127 Matrices as transformations of the plane https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:matrices-as-transformations/e/transformation-matrices-1 2026-04-08T00:36:09.500768563Z Understand matrices as transformations of the plane Use your understanding of how matrices transform the plane to solve various problems. Sulinya exercise https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:matrices-as-transformations/v/proof-matrix-determinant-gives-area-of-image-of-unit-square-under-mapping 2022-10-07T00:22:18.752150461Z Proof: Matrix determinant gives area of image of unit square under mapping Proof that the determinant of a 2X2 matrix gives the area of the image of the unit square under the mapping defined by the matrix. Sal Khan video https://cdn.kastatic.org/ka_thumbnails_cache/6347e48f-fcbb-4bcb-9959-d4cb4c91eb5a_1280_720_base.png Proof: Matrix determinant gives area of image of unit square under mapping Proof that the determinant of a 2X2 matrix gives the area of the image of the unit square under the mapping defined by the matrix. https://cdn.kastatic.org/ka-youtube-converted/_OiMiQGKvvc.mp4/_OiMiQGKvvc.mp4 259 Matrices as transformations of the plane https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:matrices-as-transformations/a/matrices-as-transformations 2021-04-29T07:59:21Z Matrices as transformations Learn how exactly 2x2 matrices act as transformations of the plane. article https://www.khanacademy.org/math/precalculus/x9e81a4f98389efdf:matrices/x9e81a4f98389efdf:matrices-as-transformations/a/practice-associating-matrices-with-transformations 2021-04-29T07:57:13Z Matrix from visual representation of transformation Learn how to determine the transformation matrix that has a given effect that is described visually. article