Identify transformation matrices

Learning Objectives Identify the transformation represented by a given 2x2 matrix. Distinguish between matrices for reflection, rotation, dilation, and shear. Determine the effect of a transformation matrix on the standard basis vectors (i-hat and j-hat). Recognize the matrix forms for common transformations (e.g., reflection across axes, rotation by 90°/180°/270°). Use the determinant of a matrix to help classify the transformation (e.g., orientation-preserving vs. orientation-reversing). Identify the scale factor from a dilation matrix. How does a video game engine instantly rotate a character or resize an object on screen? 🎮 It's all done with the power of transformation matrices! In this tutorial, you'll learn the secret code behind geometric transformations....

TermDefinitionExample Transformation MatrixA matrix that, when multiplied by a coordinate vector, produces a new vector representing the transformed coordinates of the original point.The matrix T = [[0, -1], [1, 0]] transforms the point (x, y) to (-y, x). Basis VectorsThe fundamental vectors that define a coordinate system. In 2D, these are i-hat (a unit vector along the x-axis) and j-hat (a unit vector along the y-axis).i-hat = [1, 0]ᵀ and j-hat = [0, 1]ᵀ. Identity Matrix (I)The matrix that represents no transformation. Multiplying any vector by the identity matrix leaves the vector unchanged.I = [[1, 0], [0, 1]]. Applying this to (x, y) results in (x, y). Dilation (Scaling)A transformation that uniformly enlarges or shrinks an object from a central point by a scale factor, k.A dilation...

The Basis Vector Rule For a transformation matrix T = [[a, c], [b, d]], the first column [a, b]ᵀ is the new position of the basis vector i-hat [1, 0]ᵀ, and the second column [c, d]ᵀ is the new position of the basis vector j-hat [0, 1]ᵀ. This is the most powerful rule for identifying any transformation. By observing where the basis vectors land, you can deduce the overall geometric effect of the matrix. Rotation Matrix (Counter-clockwise) R(θ) = [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]] This is the general form for a matrix that rotates points counter-clockwise by an angle θ around the origin. To identify a rotation, check if the matrix fits this pattern for a specific angle. Dilation Matrix D(k) = [[k, 0], [0, k]] This matrix scales both the x and y coordinates by a fact...