Matrix Transformation on a Vector?

📌 CONCEPT: A linear transformation represented by a matrix transforms a vector into a new vector by multiplying the matrix with the vector, resulting in a new vector that may be a scalar multiple of the original vector, a reflection, or a rotation, depending on the matrix and vector involved.

📐 RULE / FORMULA: The linear transformation is given by Av, where A is the matrix representing the transformation and v is the vector being transformed, resulting in a new vector that can be described geometrically.

💡 WORKED EXAMPLE: Let's consider the given matrix A = [[2, 1], [4, -3]] and vector v = [7, -5]. To find the new vector Av, we perform matrix multiplication: Av = [[2, 1], [4, -3]] * [7, -5] = [16, -11]. The new vector Av = [16, -11] can be obtained by multiplying the original vector v = [7, -5] by a scalar factor, indicating that the transformation is a scaling of the original vector.

⚠️ COMMON MISTAKE: Students often forget to perform the matrix multiplication correctly, which can lead to incorrect results. They should carefully follow the rules of matrix multiplication to avoid this common mistake.