Consider the transformation from to . Is T a linear transformation? If so, find the image, rank, kernel, and nullity of T.
the solution is
Yes,T is a linear transformation.
Since T satisfy
is a linear transformation.
Kernel of T:
Image of T:
Yes, is a linear transformation.
We say that an matrix A is triangulizable if is similar to an upper triangular matrix B.
a. Give an example of a matrix with real entries that fails to be triangulizable over R .
b. Show that any matrix with complex entries is triangulizable over C . Hint: Give a proof by induction analogous to the proof of Theorem 8.1.1.
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