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General Instructions: Show your work and justify all answers. You may use rref in Matlab for any problem. Write the reduced row echelon form of the matrix computed by Matlab. 1. (2 pts per part) Determine which sets are bases for R² or IR3. Justify your answers. 3 9 2 a. b. 2 , 2 c. 1 -5 , 7 1 1 2. Consider the coefficient matrix, 3 6 -9 3 9 1 2 0 -5 1 A = 4 8 -9 -2 7 -2 -4 5 0 -6 a. (3 pts) Determine a basis for the null space of A. or explain why a basis does not exist. b. (3 pts) Determine a basis for the column space of A. c. (2 pts) What is the dimension of Nul(A)? What is the rank of A? 3. Consider the coefficient matrix. -3 3 1 5 A= 2 1 2 4 1 1 a. (3 pts) Determine a basis for the null space of A. or explain why a basis does not exist. b. (3 pts) Determine a basis for the column space of A. c. (2 pts) What is the dimension of Vul(A)? What is the rank of A? 4. (3 pts) What is the rank of a 15 x 20 matrix whose null space is 4-dimensional? 5. (3 pts) Construct a 4 x 3 matrix with rank 2. 6. (3 pts) Suppose A = uvT. where u an are N x 1 vectors with at least one nonzero. Show that rank(A) = 1. 7. (3 pts) Let A E R™Xn with m > n. Select all of the following statements that are equivalent to the statement. rank(A) = n. (You don't need to explain.) i. dim(Nul A)=0 ii. dim(Nul A) = m n iii. The columns of A are a basis for IR". iv. The columns of A are linearly independent. v. For any b € R™, Ax = b has a unique solution. vi. For any b E R", Ax = b has at most one solution. vii. For any b € R", Ax = b has infinitely many solutions. 8. (3 pts) Let A € Rmxn with m to the statement, rank( (A) = m. (You don't need to explain.) i. im(Nul A)=0 ii. dim(Nul A) = n m iii. The columns of A are a basis for IRm iv. The columns of A span R™. v. The columns of A are linearly independent. vi. For any b € R™, Ax = b has a unique solution. vii. For any b € R", Ax = b has at most one solution. viii. For any b € R™, Ax = b has infinitely many solutions.

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