Solutions - UCR Math Dept.
... 5. Let V be a vector space over a field F with basis {u, v}. If a ∈ F and a 6= 0, is {u + v, av} a basis for V ? Why or why not? {u + v, av} is a basis for V. Let S = {u + v, av}, B = {u, v}. In order for S to be a basis for V , by definition for a basis, the following must be true. (a) SPAN(S) = V ...
... 5. Let V be a vector space over a field F with basis {u, v}. If a ∈ F and a 6= 0, is {u + v, av} a basis for V ? Why or why not? {u + v, av} is a basis for V. Let S = {u + v, av}, B = {u, v}. In order for S to be a basis for V , by definition for a basis, the following must be true. (a) SPAN(S) = V ...
Addition of polynomials Multiplication of polynomials
... Since we also have the Division Algorithm in R[x], we can use a similar process to find greatest common divisors in R[x]. Example 7. Find the greatest common divisor of a(x) = 2x3 +x2 −2x−1 and b(x) = x3 −x2 +2x−2. Solution. We use the Euclidean Algorithm: first divide a(x) by b(x), then divide b(x ...
... Since we also have the Division Algorithm in R[x], we can use a similar process to find greatest common divisors in R[x]. Example 7. Find the greatest common divisor of a(x) = 2x3 +x2 −2x−1 and b(x) = x3 −x2 +2x−2. Solution. We use the Euclidean Algorithm: first divide a(x) by b(x), then divide b(x ...
from scratch series........... Maximal Ideal Theorem The quotient of a
... are called left or right ideals. Note that Rotman’s terminology is complicated by his decreeing that all rings in Chapter 3 shall be commutative and have an identity. Then ideals cannot be subrings, by Rotman’s provisional definition, since once a unit gets into an ideal, it takes over the whole rin ...
... are called left or right ideals. Note that Rotman’s terminology is complicated by his decreeing that all rings in Chapter 3 shall be commutative and have an identity. Then ideals cannot be subrings, by Rotman’s provisional definition, since once a unit gets into an ideal, it takes over the whole rin ...
17-Distribution Combine Like terms
... The leading term is 3x6. There are five terms The leading term is 3x6 the exponent of the first term is 6. The leading coefficient is 6. This is a sixth-degree polynomial. ...
... The leading term is 3x6. There are five terms The leading term is 3x6 the exponent of the first term is 6. The leading coefficient is 6. This is a sixth-degree polynomial. ...
Principal Ideal Domains
... (1) a is said to be multiple of b if there exists an element x ∈ R with a = bx. In this case, b is said to divide a or be a divisor of a, written b | a. (2) A greatest common divisor of a and b is a nonzero element d such that (a) d | a and d | b, and (b) if d 0 | a and d 0 | b, then d 0 | d. A grea ...
... (1) a is said to be multiple of b if there exists an element x ∈ R with a = bx. In this case, b is said to divide a or be a divisor of a, written b | a. (2) A greatest common divisor of a and b is a nonzero element d such that (a) d | a and d | b, and (b) if d 0 | a and d 0 | b, then d 0 | d. A grea ...
