Let V = R4 . Find an orthogonal basis for the subspace spanned by e1 +e2 −e3 + 2e4, e1 +e2 + e3 + 2e4, e1 + e2 + 2e4, e1 + 2e2 + e3 + 2e4, e1 + 2e4, from this set of vectors.

Q: Let W be a subspace of the vector space V. Prove that the zero vector in V is also the zero vector…

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Q: Find a basis for the subspace S of R4 consisting of all vectors of the form (a + b, a − b + 2c, b,…

A: We have to find the basis for the subspace S. We have to find the dimension of S.

Q: Let R have the Euclidean inner product, and let W be the subspace spanned by the vectors uj = (1, 0,…

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Q: Let V be a vector space over a field F and let WC V. Show that W is a subspace if and only if W is…

A: Given: V be a vector space over a field F and W⊆V. To show: W is a subspace if and only if W is not…

Q: Let W be the subspace of R3 spanned by the following five vectors: u1=(0, 1, 0), u2=(1, 2, 0),…

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Q: 2. Let W be an (n – 1)-dimensional subspace of V. Show that V has a basis B satisfying Bn W = Ø.

A: Given W is an n-1 dimensional subspace of V which is n-dimensional. To show V has a basis B…

Q: Let S = {i, j} is a basis of R² and let W = {(x, y), x + y = 0} a- Show that W is subspace of R², b-…

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Q: Use the inner product (5, 9) = | f(=)g(=) dz in the vector space P(R) of polynomials to find the…

A: Givenf(x)=3x2+1g(x) = x h(x) = 1We know that the orthogonal projection of f onto the subspace…

Q: Let H be the set of all vectors of the form (a – 3b, b – a, a, b) where a and b are arbitrary…

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Q: Let V = Poly2 be the vector space of polynomials of degree 2 or less and let W = Poly, be the vector…

A: Suppose we have given a linear transformation ,   is basis of V ,  is basis of W  and associated…

Q: Use the inner product (5, 9) = [ f(x)g(x) dæ in the vector space P(R) of polynomials to find the…

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Q: Let W be the subspace of P3 spanned by {t³ +t² – 2t + 1, t² + 1, t³ – 2t, 2t³ + 3t² – 4t + 3}. Find…

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Q: 2. Let W be an (n − 1)-dimensional subspace of V. Show that V has a basis B satisfying Bn W = Ø.

A: Please find the answer in next step

Q: Prove that the set of vectors of the form (a, b, c) where b = a +c is a subspace of R³.

A: According to the given information, it is required to prove that:

Q: Let W = {a + bx + cx? + dx³ € Pg : c = a – b+ 2d} be a subspace of P3. Then a basis for W is:

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Q: Recall that B = {1+2x,3 – x}. Let the vector space P¡ have the inner product (p, q) = 2aobo + 3a¡b¡…

A: Given B=1+2x,3-x is basis of the vector space P1. The inner product , defined on P1 as follows:…

Q: and let W the subspace of R" spanned by i and v. Find a basis of W-, the orthogonal complement of W…

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Q: If W is a subspace of R" of dimension k with a basis {v1,e,2,..., v%} then W Range(|v1 v2 vn)). ...…

A: Given: Let W be a subspace of Rn and dimk=k Let B be a basis of k then B=k Also B is a linearly…

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: Suppose the set ​{u​, v​} forms a basis for a subspace W in Rn.  Let x​=u​+v and y​=u​-v.  Show that…

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Q: Let W be the subspace of R4 spanned by {u₁ = (1, 1, 1, 1), u2 = (2, 4, 1, 5), u3 (2, 0, 4,0)). Using…

A: The solution is given as

Q: Consider the vector space V = C2 with scalar multiplication over the real numbers R and let W and U…

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Q: Let W be a nonempty subset of a vector space V . Prove that W is a subspace of V if and only if r u…

A: Given: Let W be a nonempty subset of a vector space V

Q: Let W1 and W2 be two subspaces of a finite dimensional vector space V over a field F. 1. Show that…

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Q: Let V be a finite-dimensional vector space and T be the projection on W along W’, where W and W’are…

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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: Let w = ( { (x.y,z) ER3: x+y-z 0) be a subspace. A basis of w+ is [A].

A: Solution

Q: Let T be a linear operator on a vector space V, and let W1,W2, . . . ,Wk be T-invariant subspaces of…

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Q: Let W be the subspace of R spanned by the vectors -2 and Find the matrix A of the orthogonal…

A: Given that  W be the subspace of ℝ2 spanned by the vectors                           1-21  and…

Q: If W is a subspace of ℝ4 whose dimension is 2 then is any set of 2 vector is a basis for W

A: Answer

Q: Is W a subspace of the vector space? W is the set of all matrices in Mn,n with zero determinants.

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Q: Show that the nonempty subset W of a vector space V is a subspace of V if and only if for every pair…

A: Consider a vector space V over a field F. Suppose that W is a non empty subset of V such that…

Q: 1. Consider the vector space V = M3x3(C) over the field R. Let U be the subspace of V consisting of…

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Q: 9

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Q: Let W be the subspace of IR' spanned by the vectors 1 4 -2 and -1 -7 Find the matrix A of the…

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Q: Suppose the set {u,v} forms a basis for a subspace w in Rn. Let x=u+v and y=u-v. Show that {x,y} is…

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Q: Let H be the set of all vectors of the form (a – 3b, b – a, a, b) where a and b are arbitrary…

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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 {u1 (x) = 12, u2 (x) = 12x, u3 (x) = 8x² } be a basis for a subspace of P2. Use the Gram-Schmidt…

A: Orthogonal basis, therefore inner product will be zero for distinct two elements.

Q: Consider the vector space P3 of polynomials of degree at most 3 together with the zero polynomial..…

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Q: Let {u1, u2,..., Un} be an orthonormal basis for a subspace S of an inner product space V, and let…

A: The solution is given below

Q: Find a basis for W, the orthogonal complement of W, if W is the subspace spanned by 2

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Q: True or False: If W is a subspace of a vector space V and B is a basis for V , then B can be…

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Q: Let V and W be the subspaces of the vector space R^4 spanned by V1= (3,-1,4,1), V2 = (5,0,5,1), V3 =…

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Q: Let uj,..., u, be an orthogonal basis for a subspace W of R", and let T : R" → R" be defined by T(x)…

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Q: Let f(x) = -1, g(x) = -7x + 2 and h(x) = 5x². %3D Consider the inner product (P, 9) = / P(=)q(=) dz…

A: Given fx=−1, gx=−7x+2 and hx=5x2 And p,q=∫04pxqxdx Gram Schmidt orthogonalization process is  ax=fx…

Q: Let {w1, w2, ..., Wk} be a basis for a subspace W of the vector space V. Let v be a vector in W1.…

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Q: Find a basis for the subspace of R° consisting of all vectors perpendicular to v, that is find a…

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Q: . Let W be the subspace of R' generated by the vectors (1, –2, 5, –3), (2, 3, 1, –4) an (3, 8, –3,…

A: Consider the metrix A with given three vector as row…

Q: Find a basis of the subspace W in R*, where W is the set of vectors whose components add to zero.

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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.