Let A be the matrix of the linear transformation T. Without writing A, find an eigenvalue of A and describe the eigenspace.
T is the transformation on R3 that rotates points about some line through the origin.
Solution
T is the transformation on R 3 that rotates points about some line through origin.Let the line be fixed.
Let (x,y,z) be a point on the line. Then any other point on it is c(x,y,z) for some real number c.
Now t is an eigen value of T means, there is a non zero (a,b,c) such that (T-tI)(a,b,c)=0.
where I is the identity transformation.
That is T(a,b,c)=t(a,b,c)
So we are looking for scalars t and vectors (a,b,c) such that above equation holds.
To be more precise, we want that (a,b,c) which are just multiplied by a constant t if T is applied on them.
In other words, we are looking (a,b,c) such that this point is just fixed or moved along some line through origin origin.
Since T is a rotation, the only points that are not moved are the points on the line about which the rotation takes place.
Also the points on the line are fixed. That is T(x,y,z)=(x,y,z) for all points (x,y,z) in the line.
So 1 is an eigen value of T and the given line(since it is not moved by T) is the eigen space corresponding to 1.
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Let A be the matrix of the linear transformation T- Without writing A-.docx
1. Let A be the matrix of the linear transformation T. Without writing A, find an eigenvalue of A
and describe the eigenspace.
T is the transformation on R3 that rotates points about some line through the origin.
Solution
T is the transformation on R 3
that rotates points about some line through origin.Let the line be
fixed.
Let (x,y,z) be a point on the line. Then any other point on it is c(x,y,z) for some real number c.
Now t is an eigen value of T means, there is a non zero (a,b,c) such that (T-tI)(a,b,c)=0.
where I is the identity transformation.
That is T(a,b,c)=t(a,b,c)
So we are looking for scalars t and vectors (a,b,c) such that above equation holds.
To be more precise, we want that (a,b,c) which are just multiplied by a constant t if T is applied
on them.
In other words, we are looking (a,b,c) such that this point is just fixed or moved along some line
through origin origin.
Since T is a rotation, the only points that are not moved are the points on the line about which the
rotation takes place.
Also the points on the line are fixed. That is T(x,y,z)=(x,y,z) for all points (x,y,z) in the line.
So 1 is an eigen value of T and the given line(since it is not moved by T) is the eigen space
corresponding to 1.
