Let L = {(x1, x2, x3, X4, X5) E R° | x1 X2 + 2x4 = 0, x2 – 3x4 = 0}. | Prove that L is a subspace of R³, determine a basis of L and calculate its dimension.

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 {u1 (x) = -3, u2 (x) = – 12x, u3 (x) = -84²} be a basis for a subspace of P2. Use the Gram-…

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Q: Let i = Find a basis of the subspace of R* consisting of all vectors perpendicular to v. 9. 6.…

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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: The set B=(v= (1,1,1),u=(0,0, - 3),w=(2,2,0)) IS: OA Linearly independent in R3. OBa subspaces of R.…

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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: Consider the vectör X1 = X2 = 4. in R3, and let W be the subspace Span{x1, X2}. (a) Find an…

A: Given :-  Consider the vectors  x1 = 2-11   , x2 = 5-44 in ℝ3. and let W be the subspace span  x1 ,…

Q: Consider the vector space V = {p(x) = ao +ajr+azx² | a¡ € R, deg(p(æ)) < 2} of polynomials of degree…

A: Let          V(x)=p(x)=a0+a1x+a2x2 | ai∈ℝ, deg(p(x))≤2 of polynomials of degree less than or equal…

Q: 3. Let U be the subspace of Rª defined by U = {(x1, 12, X3, X4) E R* : ¤1 = –-L2 and 13 = 2x4}. %3D…

A: U is a subspace of ℝ4 defined as U={(x1,x2,x3,x4)∈ℝ4: x1=-x2  and x3=2x4} a) We need  to find 2…

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: Let W {(a,b, c, d)| 5a - c = 0 } be a subspace of R*. Then a basis for W is

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Q: Consider the subset W = { a+bx+cx ² = b = c } ≤ R₂ [x] A) Prove that Wis a subspace of R₂ [x] B)…

A: Vector space and it's subspace

Q: Let -1 2 4 and let W the subspace of R spanned by i and v. Find a basis of W-, the orthogonal…

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Q: The vector x is in a subspace H with a basis B= {b, ,b2}. Find the B-coordinate vector of x. x= [x]B

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Q: Find an orthogonal basis of the subspace of spanned by the vectors (a) (b) -2 (c) (d) 234

A: To find the orthogonal basis of the subspace of spanned by the vectors 234 and 123 Consider that the…

Q: Let W be the set of all vectors = E R such that xı – 2x2 + 3r3 = 0. (Geometrically, W is a plane…

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Q: 3. Let V be a subspace of R" and B:= {v1,..., V} be a basis for V. Prove that each vector in V can…

A: Explanation of the answer is as follows

Q: Find an orthogonal basis for the subspace H of IR that is spanned by the vectors 1 1 1 -2 1 V1 = V2…

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Q: Let V be the vector space of 2 × 2 matrices over the real field R. Find a basis and dimension of the…

A: Let us consider the vector space V; the set B is the basis of vector space V if  B is linearly…

Q: 0-0- Let ui = u2 = and set W = {w e Rª : w · v1 = w · v2 = 0}. W is a subspace of R4. Find a basis…

A: Find the dimension first and find vectors which satisfies given two relation

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: 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 V be the subspace of ℝ4 spanned by the columns of the matrix  A=[1 2 3 −2 3 8 5 1 −3 −3 −4 −5 ]…

A: (a)Consider the given matrix, the columns of which spanned V:

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: Let U be the subspace of R' defined by U = {(x1,22, 23, 74, #3) E R³ : #1 = 3x, and z3 = 774} . (a)…

A: Let U be the subspace of R5 which is defined by U=x1,x2,x3,x4,x5∈R5:x1=3x2 and x3=7x4 (a) We have to…

Q: Find a basis for the given subspace of R°, and state its dimension for the plane 3 x- 2 y+ 5 z = 0.…

A: We have the vector space R3 , the subspace here is: W={(x,y,z) | 3x-2y+5z=0 ; x, y, z∈R}

Q: Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned…

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Q: Let 5 Find a basis of the subspace of R4 consisting of all vectors perpendicular to v.

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Q: Find a basis of a subspace U in R given by a system: I1 - T2 + 2r3+T1 T1- 12 + 313 + 2r, 0

A: (a) Given the system x1-x2+2x3+x4=0x1-x2+3x3+2x4=0 which describes the subspace U∈ℝ4.   To find the…

Q: Let f(x) = 4, g(x) = 4x + 6 and h(x) = 2x². Consider the inner product (p, q) = p(x)q(x) dz in the…

A: The given question can be solved as shown in step2.

Q: Let 3 -7 ,V = u -3 -3 4 and let W the subspace of R* spanned by u and v. Find a basis of w', the…

A: The given vectors are u=3-73-3, v=-12-34 and W be the subspace of R4 spanned by u and v.

Q: 2. Show that {1,x – 1, x (x + 1)} is a basis of the space of polynomials P2 (IR) and that W = {p(x)…

A: If the elements of the set are linearly independent then the set is a basis set. If Wvis a 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 W = {(a, b, c, d) E R*| a – b = 0,2b + c = d}. i. Show that W is a subspace of R*. ii. Find a…

A: We will prove the following result.

Q: 2 Let A = | -4 2 Define U = {M E M2x2 such that MA = 0}. Prove that U is a subspace of M2x2. Find a…

A: We will find out the required solution.

Q: 3) Suppose that U and W are subspaces of a vector space V with V = U €W. Suppose also that u, ., Um…

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Q: Consider the vector space V = C° with scalar multiplication over the real numbers R and let W- (z1,…

A: Given is a space W in V. The objective is to prove that W is a subspace of V.

Q: Let U={p E p2 : p(2)=0}.Show that U is a subspace of p2 and find a basis for U.

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Q: Let H={p € P3: p(0)=p(1)%3D0} be a subspace of P3. Show that 8 = {-2+2t, - is basis for H using…

A: Given: H={p∈ℙ3:p(0)=p(1)=0} is a subspace of ℙ3 B={-2t2+2t,t3-t} To show that B is a basis for H…

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=R2 and let H be the subset of V of all points on the line 2x+3y=6. Is H a subspace of the…

A: Given : V=ℝ2 and H=x,y/2x+3y=6 We want to check is H a subspace of the vector space V

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: (b) Consider the vector space V = {p(z) = ao + a1r+ azx² | a; € R, deg(p(x)) < 2} of polynomials of…

A: Consider the vector space V=px=a0+a1x+a2x2: ai∈ℝ, degpx≤2 and W=px∈V:p(1)=0

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 V be the vector space of 2 × 2 matrices over the real field R. Find a basis and 5 dimension of…

A: Given A=12-13B=251-1C=51211D=34-25

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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Q: Find an orthogonal basis of the subspace of spanned by the vectors 目 目 [2] (a) (b) 3 (c) (d) 12 2 3…

A: We have to find an orthogonal basis of the subspace.

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.