Given a vector \( v = [2, 3] \) and a transformation matrix \( T = [[1, 0], [0, -1]] \), what is the projection of \( v \) using the transformation matrix \( T \)?
Answer
To perform the matrix-vector multiplication, we multiply the elements of each row of the matrix with the corresponding elements of the vector and sum them up. This gives us \( [1*2 + 0*3, 0*2 + (-1)*3] \) which simplifies to \( [2, -3] \).
Step 1 ：The projection of a vector \( v \) onto a transformation matrix \( T \) is given by the formula \( T \cdot v \).
Step 2 ：Substituting the given values, we get \( T \cdot v = [[1, 0], [0, -1]] \cdot [2, 3] \).
Step 3 ：To perform the matrix-vector multiplication, we multiply the elements of each row of the matrix with the corresponding elements of the vector and sum them up. This gives us \( [1*2 + 0*3, 0*2 + (-1)*3] \) which simplifies to \( [2, -3] \).
