Find Ai and Aj in terms of i and j.
If v is any vector xi+yj, then Av is xAi+yAj.
Note
Ai =cosσi +sinσj
Aj = your turn
Hi again everyone!
I have another question:
Let T : R^{2} -> R^{2} be the transformation that rotates each point in R^{2} about the origin through an angle σ, with counterclockwise rotation for a positive angle. Find the standard matrix A of this transformation.
I understand that a standard matrix for a linear transformation T is the matrix relative to a standard basis, but for some reason I feel like I don't have enough information in this exercise to find it?
Thanks again for taking a look at this, you people on this forum are really the best
As Hartlw suggests, you can find the matrix representation of a linear transformation by determining what the transformation does to each basis vector. In a given basis, the basis vectors themselves are represented by and . And, of course,
and .
That is, the columns of the matrix representation of a linear transformation, in a given basis, are the vectors the linear transformation maps those basis vectors into.
Now, what does a rotation, through degrees maps (1, 0) into a point (x, y) at distance 1 from (0, 0). If you drop a vertical to the x-axis you have a right triangle with legs of length "x" and "y", hypotenuse of length 1, and with angle . Then and . That is, the rotation maps the (1, 0) into . Similarly, the point (0, 1) is mapped into
Therefore the matrix representation of that rotation is ???????
To transform the point to by rotating by an angle of anticlockwise, first note that we can write and , and so after the rotation we have and . Simplifying these using the angle sum identities we find
and
Therefore when we write these in the matrix form we have
So the transformation matrix is .
Wow, thank you everyone I worked through all the answers and now I feel like I finally understand what they mean when they say "find the standard matrix of linear transformation", I finally understand that it is all about seeing what the transformation does to the basis vectors. I think I did not properly understand that when I asked the question.
Also, I liked the alternative way of solving the exercise that "prove it" showed.
Thank you all for the help! I really appreciate it!