Let R" be a real vector space, and u = (u1, u2, Un), v = (v1, V2, ..., Vn) E .... R", then show that < u, v >= U1.V1 + u2.V2 + Un.Vn ... defines an inner product on R".

Q: Mark each of the following statements true or false:

A: Given,u=4v=5, andu,v=2,   thenu+v=5

Q: Let V be an inner product space, and let u, v E V be unit vectors. It possible that (u, v) > 1? а.…

A: Let V be an inner product space

Q: Suppose that the set A ={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary n…

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Q: Let V = R?, defined the addition by: (u1 +u2) + (V1 +v2)= ( U1 +V1 , U2 +V2 + 1) With standard…

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Q: Let {u, v, w} be a linearly independent set of vectors in a vector space V.           (a) Is {u + v,…

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Q: 2. Let R3 be a real vector space, and u = (u1, u2, u3), v = (V1, v2, v3) E R", then show that =…

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Q: Suppose that u, v, and w are vectors in an inner product space such that (u, v) = 1, (u, w) = 5, (v,…

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Q: Let V = R2, defined the addition by: (u1 +u2) + (V1 + v2)= (U1 +v1 , Uz +V2 + 1) With standard…

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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: Let V be a vector space over R and W be an inner product space over R with inner product (, ),. If T…

A: To prove this, prove that it is an inner product assuming T as one-one. Now prove the reverse order…

Q: Consider the vector space P, of all polynomials of degree = a, a,+b, b2 , then the inner-product of…

A: Given polynomials are: p(x)=a1+b1xq(x)=a2+b2(x) Inner product function is defined as:…

Q: Consider the vector space Rn with inner product{x, y} = xTy. Show that for any n × n matrix A,(a)…

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Q: Find a bases for the vector space V spanned by vector w = (1,1,0),w, =(0,1,1), w; =(2,3,1) and w,…

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Q: Prove that if u and v are vectors in an inner product space such that ||u|| ≤ 1 and ||v|| ≤ 1, then…

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Q: 2. Let V be a vector space with a positive definite scalar product. Let v1, v2, . , Vn E V be…

A: Linearly independent vectors

Q: Let S be the space of all complex vectors (x, y, z) such that y = z². Determine whether or not S is…

A: In the given problem we verify the properties of vector addition

Q: Let u, v, and w be vectors in an inner product space V and let c be a scalar   &lt;u, cv&gt;=…

A: Let u, v and w be vectors in an inner product space. Let c be a scalar. To show, &lt;u, cv&gt;=…

Q: Show that the SPAN {1, cos t, cos 2t} is a vector space over R

A:

Q: Let S={x | x=a+b +c 1 where a,b,c,deR. Show explicitly that S is a vector space.

A: We will construct a transformation matrix whose span will be the set S, and the  we will show that…

Q: In a vector space V , prove that 0v = 0 for all v ∈ V

A: In this question, we want to prove that for a vector space V , 0v=0 for all v in V.

Q: Of the following vector, prove that the mapping is an inner product for all vectors v = Vx, Vy and u…

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Q: Let V be set of all positive real number with operations uÐv = uv for all u, v e V and cou = u° for…

A: We have V is set of positive real numbers.  We have: u⊕v=uv   for all u,v∈Vc⊙u=uc  for any scalar c…

Q: Let V be a vector space. Prove the following cancellation property: If c + 0 is a scalar and cx =…

A: Given: In a vector space V, cx=cy, c≠0 To prove: x=y

Q: Show that the field R(i) as a vector space over R has a basis { 1 , i } and is isomorphic to  R_2,…

A: The field ℝ (i) is defined as ℝ (i) = {a + bi| a, b ϵ ℝ}.That implies, ℝ (i) = {a × 1+ b × i | a, b…

Q: Let V be an inner product space. Show that ||cu|| |c| ||u|| for any vector u and any scalar c.

A: Given:          V is an inner product space.   To show :            cu = c u for any vector u and…

Q: Let V = R2, defined the addition by: (U1. Uz) + (V1, v2)= (U1 +V1 , uz +vz + 1) With standard scalar…

A:

Q: Show that || || :V → R : v ||v|| is continuous at each point of V, where V is a normed vector space.

A:

Q: Let V= R2, defined the addition by: (u1 tu2) + (V1 + v2)=( U1 +v1 , u2 +v2 + 1) With standard scalar…

A: We have to show that V is form vector space or not

Q: Let V be a vector space, u a vector in V, and c a scalar. Prove 0u= 0

A: Given, Let V be a vector space, u a vector in V, and c a scalar

Q: Let V = R², defined the addition by: (U1 7 Uz) + (V1 7 vz)= ( U1 +V1 , Uz +Vz + 1) With standard…

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Q: find the scalar projection of v onto w. Also find the vectorprojection of v onto w. v = (2,−3)T , w…

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Q: Let a E R". Prove that ||x|| = max |r;| is a vector norm by showing that it satisfies i=1,...n the…

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Q: find the scalar projection of v onto w. Also find the vectorprojection of v onto w. v = (4, 1)T , w…

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Q: Check whether or not the set V is vector space over R' with respect to the indicated operations. (x,…

A: The solution is given as follows

Q: (a) In the geometry of this unusual inner product space (where ||x|| = /{x, x)), find the vector…

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Q: Let V be the first quadrant in the xy-plane; that is, let 0, a. If u and v are in V, is u + v in V ?…

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Q: (b). Let V = R? = {(r,,1)| plication. Is it a vector space? IER} with usual addition and scalar…

A: Is it vector space

Q: Let X be the vector space of all or Can we obtain the norm defined on from an inner product?

A:

Q: (ii) Prove that W is a vector space of R°.

A:

Q: Let V = R?, defined the addition by: %3D (u1 +u2) + (v1 + v2)= (U1 +V1 , U2 +V2 + 1) With standard…

A:

Q: Let u and v be vectors in an inner product space V. If ((u − v), (u + v)) = 0, show that u=v.

A: Given, u and v be vectors in an inner product space V. ((u − v), (u + v)) = 0

Q: Let V be a real vector space with inner product 〈,〉. Prove that for any u,v,w ∈V 〈u,v〉〈u,w〉≤…

A: Given Information, If V is a real vector space with inner product &lt;,&gt; then we  have to prove…

Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B' A) Given the…

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Q: The set R2 is a vector space over R. How Prove it ?

A: We need to check each and every axiom of a vector space to know that it is in fact a vector…

Q: Let T: U→V be a linear transformation from a vector space U (F) into a vector space V(F) and U(F) be…

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Q: Show that (x, y) = 4x1y1-7x2Y2 for vectors x = (x1, x2) and y = (V1 Y2) in R2 defines an inner…

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Q: Let U be the set of all vectors in R3 that are perpendicular to a vector u.  Prove that U is a…

A: We need to so for arbitrary two vectors of U linear combination is also lies in U

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.