1.. Let-V-be-setof-all-ordered-pairs of-real numbers and-consider•the-following addition and-scalar-multiplication-operations on u=(u1,u2)and v=(V1,V2):. *u+v="(U1 +V1 + 1,u2 + V2 +1), ku="(ku̟ ,'ku2).. a. Compute u+v•and-ku-for-u= (0,5), v= (-1,3)'and-k=5.. b. Show-that-(-1,-1)-=0. c. - Show that-k (u+v)•=ku+kv-and (k+m) u=ku+mu-fails-to-hold.

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A: To prove this If j and k are positive integers,

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A: (a) u + v =(-1,2)+(3, 4) = (-1+3, 2+4)= (2, 6)ku = 3(-1,2) = (0,3 x 2) = (0,6) (b) The result of the…

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Q: Let V be the set of all ordered pairs of real numbers, and consider the following addition and…

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Q: Let Z denote the set of all integers with addition defined in the usual way and define scalar…

A: We shall use the axiom A6 and A7 which contain the scalar multiplication to show this.

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Q: 4. Let ƒ,g be C² scalar fields on R³. Verify that V x (ƒVg) = Vƒ × Vg. V × (ƒVg+gVf). V· (SVg) =…

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Q: Let V be the set of all positive real numbers Determine whether V is a vector space with the given…

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Q: 9. Let V be the set of all ordered pairs of real numbers, and consider the following addition and…

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Q: 4. Let V be the set of all pairs (r, y) of real numbers, and let F be the field of real numbers.…

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Q: Show that the function defines an inner product on R3, where u = (u1, u2, u3) and v = (v1, v2,…

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A: The Euclidean norm of a matrix is given as:

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Q: Show that the function defines an inner product on R2, where u = (u1, u2) and v = (v1, v2). ⟨u, v⟩ =…

A: Given function u,v=u1v1+9u2v2 where u=u1,u2 and v=v1,v2 The function is defined on R2. To show that…

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.