00:01
Hello, we are given that t is a linear operator from r2 to r2 on r2 which takes the vector v to its projection onto the vector w onto the subspace spanned by the vector w to or it is the notation is v maps to proj of v w this vector now w is given to be this vector 3 ,1.
00:31
First we have to find the matrix of the linear operator with respect to standard basis.
00:38
Second we have to find using this matrix and that v with respect to standard basis has this coordinates what are the coordinates of tv with respect to the standard basis.
00:52
Okay, so first note that proj v w is defined to be v dot w divided by w dot w times w.
01:11
This is a scalar this is v dot w dot product of two vectors this is w dot w again dot product of two vectors so this is a scalar times a vector so this whole thing is a vector.
01:24
So let us find out what is this now suppose yeah now with respect to standard basis now we will calculate the suppose the coordinates of v with respect to standard basis are v1, v2 is v1, v2 transpose.
01:57
Suppose the coordinates of v with respect to standard basis v1, v2 and the coordinates of w with respect to standard basis it is given that the coordinates of w with respect to standard basis is 3 ,1 it is given.
02:16
Now v dot w in this case is equal to since the standard basis is orthogonal dot product in the standard basis v dot w is equal to v1 dot w1 plus v2 this is equal to v1 dot w1 plus v2 dot w2 divided by w1 square plus w2 square this scalar times the vector w.
02:50
So v1 w1 is 3 and v2 and w2 is 1 divided by w1 square plus w2 square is 3 square plus 1 square which is 9 plus 1 10 and this thing is the vector w is 3 ,1...
