Final answer:
To show that multiplication by matrix A maps each point in the plane onto the line y=2x, we need to show that for any point (x, y) in the plane, the product of the matrix A and the column vector containing the coordinates of the point (x, y) will result in a column vector [2x, 2y].
Step-by-step explanation:
To show that multiplication by matrix A=[[3,1],[6,2]] maps each point in the plane onto the line y=2x, we need to show that for any point (x, y) in the plane, the product of the matrix A and the column vector containing the coordinates of the point (x, y) will result in a column vector [2x, 2x].
Let's take an arbitrary point (x, y) and compute the product:
[3 1] * [x] = [2x]
[6 2] * [y] = [2y]
Therefore, the multiplication of matrix A by any point (x, y) results in the column vector [2x, 2y]. Since the y-coordinate is always equal to 2 times the x-coordinate, we can conclude that matrix A maps each point in the plane onto the line y=2x