Let V be a vector space over R and W be an inner product space over R with inner product (, ),. If T : V → W is linear, prove that (x, y), = (T (x),T (y)), defines an inner product on V if and only if T is one-to-one. %3D

Q: A vector y in Rn can also be viewed as an n × 1matrix Y = (y). Show that ||Y||2 = ||y||2

A:

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

A:

Q: Set V = {(x, y) | x ER, YER}. Define addition of vectors by the rule (x, y) + (x, y) = (y + y',x+x')…

A:

Q: Let V be a vector space, T a linear operator on V, and v, w E V. Suppose that T(v + w) = v – 2w and…

A:

Q: 6. Let V be a vector space over F and let u, v, we V, a, Be F. Which of the following is not a…

A: In this question, we will discuss about the properties of vector space.

Q: Show that (x, y) = 5x1y1 – 9x2y2 for vectors x = (x1, x2) and y = (y1, y2) in R2 defines an inner…

A: This question is about application of linear algebra

Q: (i) Consider the vector space M2×2 with the trace inner product ( A,B) = trace(A'B) and let A = .…

A: We are entitled to solve only one question at a time. Given - To find -

Q: Let S and T be linear maps of V on W. We define S + T as (S + T)v = S(v) + T(v) for all v ∈ V and we…

A:

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

A:

Q: ar Transformations y (7) + rank (T) = dim (V) in this case, namely,

A:

Q: Let V be an n-dimensional vector space. Define f: V→ R by f(v) = |v|". Let p,q E V.Find d fp(q).

A:

Q: Let T : V → V be a linear operator over a finite dimensional vector space V . Prove that dim(ker(T))…

A:

Q: 3) Let V be a vector space and denote Vk: := V × ... × V as the cartesian product of itself k times.…

A:

Q: Create a vector of from F(x,y,z) such that the x, y, &z components contain at least two variables…

A: Given : vector of the form F(x , y ,z)  To Find : vector of the form F(x , y ,z)  such that the x, y…

Q: If A(r, s, t) be a function of the coordinates in n-dimensional space such that for an arbitrary…

A: Given: Ar,s,t is a function of the coordinates in n-dimensional space such that for an arbitrary…

Q: 8. Let V={(x,y)x,y € R}. Define addition on V as follows: (x»x)+(x,y)=(x +2x, 1,y+Y,+2) Find the…

A: To find zero vector of V

Q: Let V, W, Z be vector spaces and let T:V-->W and U:W-->Z be linear transformations. Show that if T…

A:

Q: 2: Show that a map T is linear if and only if T(Au + v) = XT(u)+ T(v) for all A ER and vectors u, v.

A: A linear transformation satisfies  T:U-V  Then T(u+v) = T(u) +T(v)          T(au) = aT(u)  For all a…

Q: Let u, v, and w be vectors in an inner product space V and let c be a scalar. Prove (u, 0) = (0, v)…

A:

Q: Let V = P2(Q), the vector space of polynomials of degree at most 2 with rational coeffi- cients,…

A:

Q: defined as Let V = {(x,y) | x, y = R}, with vector addition and scalar multiplication (x₁, y₁) (x2,…

A:

Q: Find ProjS(u) where S is the space spanned by v1= (1/3)(2,1,2) and v2= (1/3)(-2,2,1). Find the…

A:

Q: Consider the operator(function) S on the vector space R1[x] given by: S(a + bx) = -a + b + (a +…

A: see my solutiuon

Q: Define T : R? → R by 7((x, y)) = x+y. Describe the kernel of and fibers of T geometrically.

A: The kernel is a line with slope -1 and passes through the origin.

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

A:

Q: 3. Let V denote the vector space of all functions f: R" → R, equipped with addition + : V x V → V…

A:

Q: Let V be a vector space and suppose there exist maps S, T = L(V, V) such that range(S) ℃ null(T).…

A: Let V be a vector space and suppose there exist maps S,T:V→V such that .We need to prove that the…

Q: 33. Let V - P3 be the vector space of polynomials of degree <3, and let p₁(x) = (2-x)² and p₂(x) =…

A:

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: Let V be a vector space over R with inner product . Prove that V is a vector space over C with inner…

A:

Q: Let V be the vector space of functions spanned by B = {b₁ = 2 sin x + 9 cos x, b₂ = 4 sin x + 7 cos…

A:

Q: (ii) Let W = span{ } v = and The orthogonal projection of v onto W is projw(v) = and the component…

A: Given : v=-43-7    and W=span{ -2-32 , -3-1-2 } Aim: to find (a) projw(v)      ,and   (b) perpw(v)

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.