h : V → W be a homomorphism. Show that the nullspace of h, N(h) = {v E V | h(T) = Õw} subspace of V.

Q: Show that V is a subspace of R³: V is the set of all (x, y, z) such that z = 2x + 3y

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Q: Prove that every line through the origin in R3 is a subspace of R3.

A: To prove that every line through the origin in R3 is a subspace of R3.

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Q: If W is an invariant subspace for T then W is invariant under every polynomial in T

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Q: Let U and W be subspaces of V over a ficld F. Show that the union U UW is a subspace if and only if…

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Q: Let T : V → W be a linear transformation from a vector space V into a vector space W. Prove that the…

A: As per our guidelines we are supposed to answer only first question. Please repost other question…

Q: 4. Assume that T E L(V) and that U1 and U2 are T-invariant subspaces of V. Prove that U1 n U2 is…

A:

Q: Let W be a finite-dimensional subspace of an inner product space V. Show that if T is the orthogonal…

A: Given:W be a finite-dimensional subspace of an inner product space VTo prove:If T is the orthogonal…

Q: If A is a normed subspace of E, proof that bd(A) is a subspace of X.

A: Please check the answer in the next step

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Q: Produce an example of an element of Co (R, R) which is not an element of the following vector…

A: Correct answer is f(x)= Sin(x) which is infinitely many times continuously differentiable but it is…

Q: Let S be the subspace of R4 spanned by x1 = (1, 0,−2, 1)T and x2 = (0, 1, 3,−2)T . Find a basis for…

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Q: 4. Use the definition of a subspace to prove that the set V:= :, y, z E R, y+ : 0 is a subspace of…

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Q: at | f = a is a subspace of RI0,1] if and

A: Two step test

Q: If Sz is a subspace of R" of dimension 3, then there cannot exist a subspace S, of R* such that Si c…

A: This staement is true.

Q: Determine the dimension of the subspace of R consisting of all XeR* such that x1 + 2x, = 0 and X3 –…

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Q: Let W be a subspace spanned by the u's, an

A: Given, W be a subspace spanned by u's, and write y as the sum of a vector in W and a vector…

Q: Let W = {a + bx + cx² + dx³| c – 3d = 0 } be a subspace of P3. Then the dimension of W is equal to

A: As given W be a subspace of P3.

Q: Suppose U and W are two-dimensional subspaces of R3. Show that U∩W≠{0}

A: We want to show that  U∩W≠{0}

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Q: Let V be finite dimensional vector space and let W c V be subspace with complimentary subspace W'.…

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Q: Let W be a subset of the vectors in R where W = X2 X4 -x3 = x, - x, }. Show that W is a X3 3. X4…

A: We will use the formal definition of a subspace of a vector space to prove that, W is a subspace of…

Q: Find an orthonormal basis for the subspace of C(R) generated by the functions {1, x, x²} with…

A: W e find an orthonormal basis for the subspace of C([0, 1]) generated by the function (1, x, x^2)

Q: If S, is a subspace of R* of dimension 3, then there cannot exist a subspace S, of R* such that Si c…

A: S1, S2 are subspaces of R4 such that S1⊂S2⊂R4 with S1≠ S2 and S2≠R4.

Q: Determine whether the set of all 3-tuples of the form (x, y, z), where z = x – 2y, form a subspace…

A: If W is a subset of a vector space V over a field F, then W is a subspace of V, only if it satisfies…

Q: Suppose V is an interior multiplication space with finite dimension and W, and W2 are subspaces of…

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Q: The set {(x,3): x is in R)} is a subspace of R2

A: The given set is x,3: x is in R. It is known that any subspace should contain identity element.

Q: Find the orthogonal projection of onto the subspace W of R' spanned by -1 -1 -1 1 1 projw (7)

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Q: If W, and W, are two subspaces of a vector space V (F) then prove that WinW, is also a subspace of V…

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Q: Show that a subset W of a vector space V is a subspace of V if and only if span[W] = W.

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Q: Consider the following linear operator; T : R - R given by T(x. y, z) (-y, x +2, z). Find a…

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Q: If S is a set of vectors in an inner product space V(F) then prove that S- is a subspace of V.

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Q: Suppose V is finite-dimensional, with dim V = n > 1. Prove that there exist 1-dimensional subspaces…

A: Let us consider that (v1, v2, ..., vn) be a basis for V, and let Ui= span(vi), for i ∈ {1, 2, ...,…

Q: Show that the given set V is closed under addition and multiplication by scalars and is therefore a…

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Q: Prove that Co is a Banach space by showing that it is a closed subspace of l

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Q: Show that a subset W of a vector space V is a subspace of V if and only if span(W) = W.

A: To show that W is a subspace of V if and only if span (W) is equal to W.Let W be the subspace of V.

Q: If x is in W, then x is orthogonal to some vector in a subspace W

A: SOLUTION-

Q: Let H be the set of all vectors of the form 3s . Find a vector v in R3 such that H = Span {v}. Why…

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Q: Let P, be the set of all polynomial functions in P(R) of the form ·+ anx", for some fixed positive…

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Q: () If S is a subspace in R that is spanned by then S is spanned by: O a. 1 and 0. 1 1 and 1 c. and…

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Q: Let A be an n x n matrix. Verify that V = {x e R" : Ax = 3x} is a subspace of R".

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Q: Let β be a basis for a subspace W of an inner product space V, and let z ∈V. Prove that z∈W⊥if and…

A: Let β be the basis for a subspace W of an inner product V, and let z € V.It is required to prove…

Q: Let T : V → V be a linear transformation. Carefully prove that the image of T is a subspace of V.

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: 3) Show that the set of all elements of the form (x + 2y,x + 3z, 2x + y + z), where x, y, z E F, is…

A: We have, (x+2y , x+3z , 2x+y+z),where x, y, z € F.

Q: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.

A: A set B is a basis of a vector space V if its elements are linearly independent and every elements…

Q: (a) Prove that the image T(V) = {u : u=T(x) for some x € V } of T is a subspace of W. |(b) If the…

A:

Q: Suppose W1 and W2 are subspaces of a vector space, V. Define W as follows: W = {w1 + W2 | W1 E W1…

A: As W1 is a subspace of V , hence (a) 0∈W1 (b) αw1+βw1'∈W1 for every w1 , w1'∈W1 As W2 is a subspace…

Q: Let S be a subspace of R" and let S be its orthogonal complement. Prove that S is also a subspace of…

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Q: let W={(x,,x,,1):x,,x, are real numbers} is W a subspace of R

A: False

Q: Let L : V → W be a linear transformation from a vector space V into a vector space W. The image of a…

A: By the definition of a vector subspace we need to show 3 properties. 1)0∈L(V1) 2)L(V1) is closed…

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.