Let n, r, s be positive integers and let v1,…,vr ∈ Rn . Also, let w1,…,ws ∈ Rn. If wi ∈ span{ v1,…,vr }, then each wi ( 1 ≤ i ≤ n) is a linear combination of v1,…,vr .
Apparently, span{ v1,…,vr } ⊆ span{ v1,…,vr, w1,…,ws }.
Further, an arbitrary vector u (say) in span{ v1,…,vr, w1,…,ws } is a linear combination of the vectors in span{ v1,…,vr }. Now, since each wi ( 1 ≤ i ≤ n) is a linear combination of v1,…,vr, hence u is a linear combination of the vectors v1,…,vr. Therefore u ∈ span{ v1,…,vr }.
Hence span{ v1,…,vr, w1,…,ws } ⊆ span{ v1,…,vr }.
Therefore, span{ v1,…,vr, w1,…,ws } = span{ v1,…,vr }.
Problem statement: Prove the following: Theorem: Let n, r, s be positive integers, and let v1, . . . , vr E Rn and wi,...
1) Let n and m be positive integers. Prove: If nm is not divisible by an integer k, then neither n norm is divisible by k. Prove by proving the contrapositive of the statement. Contrapositive of the statement:_ Proof: Direct proof of the contrapositive
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Abstract Algebra (6) Let R be a commutative ring. For elements r, s є R, prove the Binomial Theorem in R: Here if n Z, r є R, we interpret nr to be the element in R which is a sum f n many r's. (6) Let R be a commutative ring. For elements r, s є R, prove the Binomial Theorem in R: Here if n Z, r є R, we interpret nr to be the element in R...
Let n be a positive integer, and let s and t be integers. Then the following hold. I need the prove for (iii) Lemma 8.1 Let n be a positive integer, and let s and t be integers. Then the following hold. (i) We have s et mod n if and only if n dividest - s. (ii) We have pris + t) = Hn (s) +Mn(t) mod n. (iii) We have Hr(st) = Hn (3) Men(t) mod n. Proof....
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In the following problem, we will work through a proof of an important theorem of arithmetic. Your job will be to read the proof carefully and answer some questions about the argument. Theorem (The Division Algorithm). For any integer n ≥ 0, and for any positive integer m, there exist integers d and r such that n = dm + r and 0 ≤ r < m. Proof: (By strong induction on the variable n.) Let m be an arbitrary...
(3) Let m,n E N. Let p(x), i -1, ..., m, be polynomials with real coefficients in the variables -(x,..., rn). Prove that pi(r) p(a) Un (r)」 is a continuously differentiable map from R" to R". (Suggestion: Use Theorem 9.21.) (3) Let m,n E N. Let p(x), i -1, ..., m, be polynomials with real coefficients in the variables -(x,..., rn). Prove that pi(r) p(a) Un (r)」 is a continuously differentiable map from R" to R". (Suggestion: Use Theorem 9.21.)
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QUESTION: PROVE THE FOLLOWING 4.3 THEOREM IN THE CASE r=1(no induction required, just use the definition of the determinants) Theorem 4.3. The determinant of an n × n matrix is a linear function of each row when the remaining rows are held fixed. That is, for 1 Sr S n, we have ar-1 ar-1 ar-1 ar+1 ar+1 ar+1 an an rt whenever k is a scalar and u, v, and each a are row vectors in F". Proof. The proof...
The symbol N denotes the nonnegative integers, that is, N= {0,1,2,3,...}. The symbol R denotes the real numbers. In each of the proofs by induction in problems (2), (3), and (4), you must explicitly state and label the goal, the predicate P(n), the base case(s), the proof of the base case(s), the statement of the inductive step, and its proof. Your proofs should have English sentences connecting and justifying the formulas. As an example of the specified format, consider the...