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The spring 2007 first midterm exam for math 232 at sfu. The exam covers topics such as systems of equations, linear independence, and matrix transformations. Students are required to solve problems related to finding the augmented matrix, reduced row echelon form, parametric solution sets, linear combinations, and determining linear independence.
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x 1 +x 2 +2x 3 − 4 x 4 = 1 x 1 +2x 2 +x 3 +x 4 = 2 2 x 1 +4x 2 +2x 3 −x 4 = 1 (a) (1 point) Write down the augmented matrix corresponding to this system.
Answer
(b) (3 points) Determine the reduced row echelon form of this matrix. Show your work. (use the back of the previous page if you need more room)
Answer
(b) (3 points) Can a set of r vectors in Rn^ ever be linearly independent if n < r? Prove your statement.
Answer
(c) (4 points) Determine if the set
is linearly independent.^ Show that your answer is correct.
M =
Answer
(b) (3 points) Prove that, for an invertible n × n matrix A, the linear transformation T : Rn^ → Rn^ defined by T (x) = Ax, is onto.