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F10HW-8 - F10 Homework 8 Section 4.1 4 A problem which is...

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F10- Homework 8. Section 4.1. 4. A problem which is almost the same is posted in the Course Documents area and was briefly shown in class. You are actually expected to check all the 8 conditions. 6. The set of non singular matrices does not contain the O (zero) matrix which is singular. Accordingly this subset cannot be a subspace. (All subspaces contain the zero vector). Problem 11. a) Let A x 1 = b and A x 2 = b , then A ( x 1 + x 2 ) = A x 1 + A x 2 = 2 b . x 1 + x 2 is a solution of A x = b if and only if 2 b = b , that is if and only if b = 0 . Just the same, A ( c x 1 ) = cA x 1 = c b and c x 1 is a solution of A x = b if and only if c b = b . Either c = 0 or not, this is possible if and only if b = 0 . b) Since V ⊂ R n , the addition and scalar multiplication in V enjoy the properties that have been proved to hold in R n . c) For (3) to make sense, the vector 0 must be a solution. But A 0 = 0 , so 0 ∈ V only if b = 0 . d) If (3) does not hold, x ∈ V , ∃ − x ∈ V , such that x + ( x ) = 0 , does not make any sense if 0 is not in V since if x and x were in V , their sum 0 would be in V .
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