2. Let R3 be a real vector space, and u = (u1, u2, u3), v = (V1, v2, v3) E R", then show that < u, v >= U1.V1 + 2 u2.V2 + U3.V3 defines an inner product on R$.

Q: The set of vectors in R? with y = -3x is a real vector space

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Q: Find one voctor in R' that spans the intersoction of U and W where U is the xy-plane-that is, U =…

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Q: 2. Let P2(R) be the vector space of polynomials over R up to degree 2. Consider B = {1+x= 2x°, -1+a…

A: given B=1+x-2x2 , -1+x+x2 , 1-x+x2B'=1-3x+x2, 1-3x-2x2 , 1-2x+3x2 b) find the coordination matrices…

Q: = Consider two bases B = {u₁, u₂} and B' = {V₁, V₂} for a certain vector space V. Assume that u₁ =…

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Q: Let V be a vector space, let c be a scalar and let u, v, w be vectors in V and ca scalar. Which of…

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Q: If V is a vector space then show that cV=V.

A: Vector space: A non-empty set V closed with two binary operations called addition and scalar…

Q: 6. (a) Let u, v, w be three linearly independent vectors in Euclidean Space R^7. Determine a value…

A: we have to find the value of for the linearly dependent and linearly independent

Q: Let (V, (, )) be an inner product space and let f, g be vectors in V. if || f|| = 4, ||| 5 and || f…

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Q: Consider the set V of all vectors of the form (x y z)T where x, y and z are real numbers. 3. Show…

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Q: Consider set V = { (x, y ), x,yER} with the following operations: addition (x1, y1) + (x2 , Y2) = (2…

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Q: 2. Determine whether the following sets with the corresponding vector addi- tion and scalar…

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Q: If V is vector space, with u = (u, , U2) and v = (v, , v2) with u +v = (u,+v, , u2+v2) and ku =…

A: u=(u1,u2) , ku=(3u1,0) k=2, u=(-1,2)

Q: 8) (True/False) Use the definition of linear independence to check whether the vectors u₁, u₂, and…

A: The given vectors are

Q: 2) Write the vector v as a sum of proj span{.…...} (v) and proj spo span{m} orthogonal set V₁,..., V…

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Q: Determine the element of V that represents the zero vector 0 (i.e., determine the “value” of 0 in V…

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Q: [5] 3 R3 of space u = ve v = 2 Express it as a linear combination of vectors VECTOR 22 - O A) -u +…

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Q: What is the dimension of the vector space S=abc detEM2x3(R): a=b=c=0}

A: Dimension of vector space is number of element in basis

Q: Let V be the space on R² where addition of the vectors u and v are defined as u+v= (2u1 +3v2, 3u2 —…

A: Here we have to find zero vector and inverse vector for given vector operation.

Q: 2. fu = and v = v1, v2, V3 > are two vectors in 3-space show k that u x v= u1 U2 U3 V1 V2 V3

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Q: Determine whether the following sets with the corresponding vector addi- tion and scalar…

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Q: Show that the function (V1, V₂) = 2w1W2 + x1x2 - Y1Y2 +2122 defined for any vectors V₁ = (x1, y1,…

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Q: Let V = R2, defined the addition by: (u1 tu2) + (v1 + vz)=( U1 +v1 , Uz tv2 + 1) With standard…

A: Given that the vector addition on V=ℝ2 is defined by           u1,u2+v1,v2=u1+v1,u2+v2+1 with…

Q: Let v1, V2, V3 and u be vectors in R". If u is orthogonal to each of v1, V2, and v3 then u is…

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Q: Let ū = [1,1,0] and w = [2, 2, 1]. Find the projeci proj,u of the vector w on ū a) [1,0, 0] b) [0,…

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Q: : Let V be the set of all ordered pairs of real numbers (u1, u). Consider the following addition and…

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Q: Let V = R' and define vector addition as (x, y, z)+(x', y', z')= (x+x',y+y',z+z') (standard…

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Q: Let V be the set of all ordered pairs of real numbers (x1, u) with u > 0. Consider the following…

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Q: Let R" be a real vector space, and u = (u1, u2, Un), v = (v1, V2, ..., Vn) E .... R", then show that…

A: For a real vector space, the inner product . , . satisfies the following properties a)  u+w , v =…

Q: 10) (6 points) Using the inner product (v, w) = 3v¡W1+5v2W2 with the vector space R?. a) Find a…

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Q: d). Let K = {(x;y) | x,y e R} with operations (x1;y1) (X2;y2) = (x, + x2 + 1; y, +y2 - 2) %3! (ax, +…

A: Given: K=x;y | x, y∈ℝ with operations x1, y1⊕x2, y2=x1+x2+1; y1+y2-2 and a⊗x1;y1=ax1+a-1; ay1-2a+2.…

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.