Final Exam — Fall 2023

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Final Exam — Fall 2023

Question 1 (12 points, 2 points each)

(a) Isomorphism

(b) Orthonormal vectors

(c) A weighted inner product

(d) Replacing a vector

(e) Change of coordinates

(f) Surjectivity

Question 2 (12 points total)

(a) Similar matrices (4 points)

(b)(i) A determinant identity (4 points)

(b)(ii) Replacing a row (4 points)

Question 3 (7 points)

Question 4 (10 points total)

(a) Matrix representation (4 points)

(b) Isomorphism (3 points)

(c) Diagonalization (3 points)

Question 5 (7 points)

Question 6 (7 points total)

(a) Dimension (3 points)

(b) Matrix-space comparison (2 points)

(c) Polynomial-space comparison (2 points)

Question 7 (8 points)

Question 8 (10 points, 2 points each)

(a) Invertibility

(b) A homogeneous solution

(c) Zero as an eigenvalue

(d) Kernel dimension

(e) Image basis

Question 9 (8 points)

Question 10 (7 points total)

(a) Membership in the span (4 points)

(b) A basis for P_2 (3 points)

Question 11 (12 points total)

(a) Compute an inner product (3 points)

(b) Test for unit norm (2 points)

(c) Distance (3 points)

(d) Orthogonality (4 points)