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What the lecture covers
A linear transformation is determined by where it sends the basis vectors. In two dimensions, those destinations form the columns of its matrix, and multiplying the matrix by a vector applies the transformation. When one transformation is followed by another, their combined effect is itself a linear transformation, represented by a composition matrix. To find that matrix, track where each basis vector ends up after both steps.
The product of two matrices captures this combined effect: apply the matrix on the right first, then the one on the left. Each column of the product is found by applying the left matrix to the corresponding column of the right matrix. This geometric interpretation makes clear that matrix multiplication is generally not commutative: reversing the order of a shear and a rotation can produce a different result. It also explains associativity. Grouping three matrices in either way preserves the same order of transformations, so the overall result is unchanged.
Key ideas
A linear transformation preserves the origin and maps evenly spaced, parallel grid lines to evenly spaced, parallel grid lines.
A transformation is determined by where it sends the basis vectors, whose destinations become the columns of its matrix.
Applying one transformation after another creates a new linear transformation called their composition.
The composition matrix has the same effect as applying the individual matrices successively to any vector.
A matrix product represents successive transformations, with the right-hand matrix applied first.
Each column of a composition matrix is obtained by applying the left matrix to the matching column of the right matrix.
The order of matrix multiplication matters because changing the order of transformations can change their combined effect.
Matrix multiplication is associative because either grouping applies the same three transformations in the same order.
Sample questions
A linear transformation in the plane sends each vector x(1,0) + y(0,1) to xT(1,0) + yT(0,1). If its matrix is written using columns, what does the first column contain?
AThe coordinates of the image of (1,0) under T
BThe coordinates of the image of every vector with x = y
CThe coordinates of the original vector (1,0) before applying T
DThe coordinates of the image of (0,1) under T
Show answer
Correct answer: A. The first column records where the first standard basis vector, (1,0), is sent; the second column records the image of (0,1).
A rotation and a shear are combined into one overall linear transformation of the plane. Which statement correctly describes the combined transformation?
AIt must be represented by two separate matrices and cannot have a matrix of its own.
BIt can be represented by a matrix only if the rotation and shear are identical.
CIt is a linear transformation in its own right and can be represented by a matrix.
DIt is not a linear transformation because it combines two different operations.
Show answer
Correct answer: C. Combining linear transformations produces another linear transformation, so the overall mapping can have its own matrix representation.