There are five problems worth a total of 100 points. Start each problem
on a new piece of paper.

(a) True/False. Let u and v be nonzero vectors from R n . If u and v are linearly dependent, then v = cu for some real scalar c.

(b) True/False. A linear system of m equations in n variables always has at least one solution when m < n.

(c) Explain. Is the matrix below in reduced row echelon form?

(a) True/False. Let A be an m × n-matrix with real coefficients and let B be an n × p-matrix with real coefficients. If Null(AB) = {0} then Null(B) = {0}.

(b) True/False. Let T : V → W be a linear transformation between vector spaces V and W and let S : W → U be a linear transformation from W to a vector space U. If T : V → W is onto, then the composition S ◦ T : V → U is onto.

(c) Explain. Let A be an n × n-matrix with Null(A) = {0}. If b is a vector from R
n
,
then does the equation Ax = b have a unique solution?

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