Q1. (a). Let V = R and define addition and scalar multiplication by: (a, b) e (c, d) = (a +2c, b+2d). %3D ko (a, b) = (ka, kb).. Determine whether V is a vector space.

Q: Find the scalar product a•b when (i) a= x+2y -z and b=2x+3y +z (ii) a=x-3y +z and b=2x+y +z

A: We have to find the scalar product a·b when (i). a=x+2y-z and b=2x+3y+z(ii). a=x-3y+z and b=2x+y+z

Q: Let V = R2, defined the addition by: (uz +u2) + (vị + vz)= (U1 +V1 , Uz +V2 + 1) With standard…

A:

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: What is the dimension of the vector space M,x3R)?

A: SOLUTION-

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

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Q: The domain D(T) is a vector space

A: Given, the domain D(T) is a vector space__________

Q: Consider two bases B= (b, b2) and C= (c, c2) for a vector space V such that b, =e, - 6cz and by =…

A: Given two bases  and  for a vector space V such that and . Also, given that and we have to find .…

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

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Q: Calculate the scalar triple product u · (v × w), where u =〈1, 1, 0〉, v =〈3, −2, 2〉, and w =〈4, −1,…

A: The vectors are, u=1,1,0v=3,-2,2w=4,-1,2 We need to find the scalar triple product u·v×w

Q: Suppose that S = {v1, v2, v3} is a linearly independent set of vectors in a vector space V. Is T =…

A: Given: S=v1, v2, v3 are linearly independent vector. Also, T=w1, w2, w3 where:…

Q: Let x and y be vectors in the R³. Suppose that x is a scalar product of 2 and the vector y. Prove…

A: x and y are vector in R3, x is scalar or dot product of 2 and y vector,  it means x = 2.y,

Q: H.W: Let V = R2, defined the addition by: (u1 +u2) + (V1 +v2)= ( U1 +V1 , U2 +V2 + 1) With standard…

A:

Q: If T: V → W is a linear map between vector spaces V and W, T is one-to-one and onto, and a ba- sis…

A: Given that T:V→W is a linear map between the vector spaces V and W. Also, T is one-to-one and onto.…

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 = {(x, y): x 20) with standard operation on R2. Is V a vector space?

A: Given set is  V={(x,y):x=>0 }

Q: . u. ): Is the set S = {r +x-1,22 - r+3,-a+3z-2}) a base for the Pa(r) vector space which is a…

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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: 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: Determine the dimension of the vector space given by -a V = | a,b E R a b

A: Given V = -aa0, -b0b | a, b ∈ℝ

Q: Let V be a vector space, and let T : V → M₂2 be linear. Let V₁, V₂ EV and suppose 2 T(v₁) = J 1…

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Q: Determine the basis that generates the following vector space by clearing the variable y: M = {(x,…

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Q: What is the dimension of the vector space P°?

A: The dimension of Pn is n+1

Q: Let V ((x,y): x 20) with standard operation on R2. Is V a vector space?

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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: Use coordinate vectors to determine whether the the space P₂ of polynomials, that is, let B = {1, t,…

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Q: Find x in R2 whose coordinate vector relative to the basis B is [x]B =

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

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

A:

Q: Consider the vector space C2 with the standard inner product. Define the linear operator T:C? → C2…

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Q: Determine whether the set B is a basis for the vector space V.

A: V=P2  ,B={ 1,1+2x+3x2)

Q: Consider the vector space P,(1)of the polynomials of degree less than or equal to 2. Then check…

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

A:

Q: Let V = {(x, y): x 20} with standard operation on R2. Is V a vector space?

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Q: Let V = {(x,y) where y= 3x+1 x ER} with scalar multiplication defined by k(x,y)=(kx,y) is a vector…

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Q: T = 1 is a vector space under standard matrix addition and scalar %3D multiplication.

A: Given a set of matrices T = abcd ∈ M2x2 | c=1 To identify if this is a vector space or not.

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

A:

Q: Let V be n-dimensional vector space over c en for any n scalars ₁, ... , E F there-

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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: H.W: Let V = R?, defined the addition by: (u1 +u2) + (V1 +v2)= ( U1 +v1 , Uz +v2 + 1) With standard…

A: Let V=R2, defined the addition by…

Q: Consider the vector space P, with the standard inner product. Calculate || p || and d(p,q) for the…

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

A:

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

A:

Q: Let 0. j =| 1 and k=|0 r Find scalars c1, c2, and c3 so that any vector u = t can be written as u =…

A: Given, u=rst,  i=100,  j=010  &  k=001

Q: B. Is C a real vector space? Explain.

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Q: Let V be a vector space with a scalar product. Show that = 0 for all v E V.

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Q: Find the vector projection of b onto a when b =2i-4j+3 k, a 3i+j+2 k.

A: Given that: b=2i-4j+3ka=3i+j+2k

Q: Calculate the scalar triple product u· (v x w), where u = (1, 1, 0), v = (2, –3, 5) and w = (4, –1,…

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Q: Let B= {b,,b2}, and C={c,,c2} be Bases for a vector space V, andsuppose b1=6c,-5c2, and b2 =3c1-2C2.…

A: I am going to solve the given problem by using some simple algebra to get the required result.

Q: Determine a scalar r such that Ax = rx, where

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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.