## Transcribed Text

I. (10pts) Find all the solutions of the following system of equations (w/o a calculator).
T1 + T2 + 3.3 = 0
2.11 + T2 + 4.33 = 1
3.11 + T2 + 5.3 = 2
II. (10pts) T + x2
(a) Show that B is a basis for the space P2 of polynomials of degree less or equal to 2.
(b) Write p(x) = 1 + 3.c + 2² as a linear combination of vectors in B.
III. (10pts) Let I : R² R² be the identity operator (i.e. I(v) = v for any U € R². Let
1
1
S = {
} and T = {
-1
D
-1
,
,
}. Find the matrix representation of
1
2
I with respect to (a) S; (b) T; (c) S and T; (d) T and S.
IV. (10pts) Let A be the matrix
Is A diagonalizable? If yes, find the
0 0 2 0
1 0 0 2
diagonal matrix similar to A.
V. (20pts) Let P2 denote the space of polynomials of degree less then or equal to 2. Let
T : P2
P2 be defined as T(f) = 2f' + f".
(a) Show that T is a linear transformation.
(b) Let S = {1,t,t2} Find the matrix representation A of T with respect to the ordered
basis S.
(c) Find a basis for the kernel of T. and a basis for the image of T.
(d) Write the characteristic polynomial of the matrix A from part (b). Find the eigenvalues
of A.
(e) For each eigenvalue in part (d) find a basis for the corresponding eigenspace.
(f) Is A diagonalizable?
VI. (16pts) For each statement below answer whether it is TRUE or FALSE.
1. If a vector V is orthogonal to vectors u and W, then it is orthogonal to each vector
in the span (u, w).
2. Any set of vectors containing the zero vector is linearly dependent.
3. det = det B.
4. If a linear transformation L : R³ R³ is onto, then it is invertible.
5. If A is a 3 X 5 matrix whose null space has dimension 2, then the equation Ax ==== 0
has infinitely many solutions.
6. Every subspace of Rn has at least one orthonormal basis.
7. If A is a 3 X 3 matrix with eigenvalues / then it is diagonalizable.
8. The determinant of an elementary matrix is always 1.

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