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Unformatted text preview: r < n
(b) r = m and r < n
(c) r < m and r = n
(d) r = m = n
( pts) Solution: Answers: (a) none or inﬁnitely many; (b) one or inﬁnitely many;
(c) none or one; (d) one. Ex. 5. Finish the following sentence. Give a short explanation for your answer.
An n × n matrix A is invertible if, and only if, the dimension of its column space C (A)
is . . .
( pts) Solution: Answer: “the dimension of C (A) is equal n.”
This is the case, since A is invertible when every column of A is its pivot column, that
is, when rank r of A is equal n. But we know, that the dimension of C (A) is equal to r. In the test, there will be also a bonus exercise.
2 MATH 441.001
Instr. K. Ciesielski
Spring 2011 NAME (print): Review for TEST # 3
Solve the following exercises. Show your work. (No credit will be given for an answer with
no supporting work shown.)
(a) Find the orthogonal completion V ⊥ of the vector space V spanned by the following
three vectors. The description of V ⊥ is understood as: (i) giving (explicitly) a basis
for V ⊥ and (ii)...
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