MATH 2210 Sample Final Exam
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MATH 22l0
SAMPLE F1NAL ExAM
1. Let A be the 2 x 2 matrix
(a) Compute the determinant, the trace and the characteristic polynomial of A.
(b) Find a relationship between the determinant, the trace and the characteristic
polynomial of A.
(c) Does the relationship you have found in part (b) work for all 2 x 2 matrices? Discuss.
2. The trαηs夕Оse of a matrix A is the matrix AT whose entry in the ith column jth row is equal to the entry in the ith row and jth column of A. The function
is a linear transformation. Find 4 linearly independent eigenvectors of T.
3. Let u and v be two vectors in Rn which are orthogonal to each other. Show that
This is known as the Pythagoras’ theorem.
4. Suppose that u1 , u2 , v1 , v2 , v3 are pairwise orthogonal vectors in Rn . Let A be the matrix whose columns are u1 , u2 and let B be the matrix whose columns are v1 , v2 , v3 . What can you say about the columns of AT B .
5. Consider the linear transformation tr : Mat3X3 (R) → R defined by the trace. What is the dimension of the kernel of this linear map?
2021-12-16