Lct Vị and V2 be subspaces of R" such that Vị C V2. Show that any vector v E R" can be uniquely written as v = a+b+c with a E Vi, be Vn V2, and c E V.

Q: 3. Decide if the followin Let V and W be vector subspace of V.

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Q: Let S = i. ii. iv. [2s- - S s,te R t Inter} -S Show that S is a subspace of R4 Find two vectors that…

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Q: Find the orthogonal projection of u on the subspace of R³ spanned by the vectors v₁ and v₂. u = (1,…

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Q: Consider the vector v = (6, 1, -3) in R. Let V be the subspace of R consisting of all vectors of the…

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Q: HConsider the vector space R° (over R) and ZcR consisting of vectors z=(z1, z2 z3), with these…

A: Given Z⊂R3, where R3 vector space.z→=(z1,z2,z3)(a) z1=0(b) z1+z2=0(c) z1+z2=1We have to check in…

Q: If u and v are vectors in R and W is a subspace of R with (u - v) belong to W, then both u and v…

A: If u and v are vectors in R3 and W is a subspace of R3 with (u-v) belong to W, then both u and v…

Q: Q. 5 If V is any vector space, show that (0} and V are subspaces of V.

A: We have that V is a vector space.  Now, V is said to be a vector space over a field F along with 2…

Q: , dim W 5. Let U, W be subspaces of a vector space V. If dim U and Un W = {0}, what is dim (U+ W)?

A: Let V be a vector space over a field k and let U, W be finite-dimensional subspaces of V.  Then…

Q: The vector x is in a subspace H with a basis B= {b, ,b,}. Find the B-coordinate vector of x. 1 -3 -…

A: We will solve a system not equations to find the B-coordinate vector. The detailed solution is as…

Q: :) Let T : U = р — 2q — 0, р, 9, r, s E R Is U a subspace of M2.2(R), if so, find its basis and…

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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: to the subspace W = Span{u1, u2} where 0. Find the distance from the vector y and u2 = -2 O 2 01

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Q: a Q#01:- Let W be the set of all vectors in R3 of the form where a and La + b! b are any real…

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Q: Find the orthogonal projection of the vector v onto the subspace S. v = |2 S =

A: projS(V) = (u1.v/u1.u1)u1 + (u2.v/u2.u2)u2

Q: R? Which vector stretches the x + y = 0 subspace of ? Why is that? a. (2,1) b. (0,0) c.(1, -1)…

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Q: Given v = find the coordinates for v in the subspace W spanned by -7 u1 = and uz -1 -28 Note that uj…

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Q: - -{}}} {[:]} (a) Consider the subspace W = span 3 and consider the vector u = Find the vector w in…

A: Given that                      W=span23-1, 102 and the vector     u=1-1-1

Q: Let-{ », :x, y, a, b e R and x - a = y – b}.Is W a subspace?

A: We will use the one step test for vector subspace to check if W is a subspace or not. We know that…

Q: Let S consists of all vectors of the form (x, 0, y, 0) in R* whe y are real numbers. Determine…

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Q: Let W = (a + bx + cx? + dx³la + 2c 0 andb-d D 0) be a subspace of P,. %3D Then dimension of W is…

A: Given  W={a+bx+cx2+dx3| a+2c=0 and b-d=0}Since a+2c=0⇒a=-2cand b-d=0⇒b=dwhich gives…

Q: Prove that if U and V are subspaces of R" and U <V then V+<U+.

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Q: Find the distance from the vector -1 y = -5 10 to the subspace W spanned by -2 1 u 0 and v= 2 ||

A: The given vector is y=-1-510 and the subspace is u=-200, v=12-1.

Q: Consider the three vectors V1 = (1,0,0, –1), v2 = (0, 1, 1, 0), V3 = (1, 1, 1, 1). Let W be the…

A: Given that W be a subspace of R4 spanned by the vectors. v1=(1,0,0,-1), v2=(0,1,1,0), v3=(1,1,1,1)…

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

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Q: Find the projection of the vector v onto the subspace S.

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Q: c) Find dimension of subspace of all vectors in R° whose first and third entries are equal.

A: We have to find the dimensions

Q: If u and v are vectors in R and W is a subspace of R with (u + V) belong to W, then both u and v…

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Q: Give a counter example to show that W is not subspace of R³. W=set of (x, y, z) where x + y = 1 6. ,

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Q: Q. Show that (a,b,c) where c=a+b is subspace of R

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Q: (b) Define subspace and Verify whether is subspace v = {(x,y,2):x,y,2ER² and x-2y+3z = 5}

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Q: (a) Determine whether the set of all vectors of the form (a,b,c), where b= a+c, is a subspace of R³.…

A: Consider the subset S of ℝ3 defined as, S=(a,b,c)∈ℝ3 | b=a+c Take  a1,a2,a3,b1,b2,b3…

Q: 3. Is the set of all vectors (x, y) in R2 with the usual addition and scalar multiplication, a…

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Q: 4) Find all values of h such that Y will be in the subspace of 3. R spanned by VI) V2, V3 if V I V2…

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Q: Be F1(x, x, x) e R³ and F2(a, b,0) e R³ a)show that F1 and F2 are subspaces of R b)show that R° = F1…

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Q: The vector x is in a subspace H with a basis B = {b, ,b,}. Find the B-coordinate vector of x. 1 - 2…

A: Given that, b1=14-2; b2=-2-73; x=15-3 Since B = b1, b2 is the basis, xB is the linear combination of…

Q: he zero vector in U is a subspace of V.

A: Inroduction a. Zero vector is a subspace of every vector space.

Q: а о if v = M22 ( R) and W = { : а , bE R } 0 a then .a W is not subspace of .b W is subspace of

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Q: Which of the following are subspaces of R^3?  W ={(x, y, z) :x+y+z=0} W= {(x,y,z): x2 + y2+ z2…

A: The given vector space is R3. We know that a subset W of a vector space V is a subspace of V if it…

Q: In R, let S be the set of all vectors with third component 0. { a S = ER Show that S is a subspace…

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Q: 10. Find the dimension of the subspace V span{v1, v2, V3, v4} in R³ where vi 1 V2 = , V3 = and v4 =

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Q: Let U and W be subspaces of R³ for which dimU=1, dimW=2, and UgW. Show that R=UOW.

A: I have shown all the conditions of direct sum of two subspaces

Q: (A) Let U1, U2, U3 be subspaces of M2x2(R). If V = U1 e U2 e U3, then dim V = 4.

A: So,  V is the direct sum of the subspaces U1, U2, U3  of M2×2(R) , i.e. dim(U1) = dim(U2) = dim(U3)…

Q: Find the orthogonal projection of u = 1 onto the subspace of R spanned by the vectors 3 2 V1 = V2 =…

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Q: Investigate the dimension of the following subspace of R*. All vectors of the form (a,b,c,d), where…

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Q: Determine: if W = the set of all vectors in R^2 whose second component is the square of the first.…

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Q: Find the orthogonal projection of v onto the subspace W spanned by the vectors ui

A: Given:   v = 123 , u1 = 2-21 , u2 = -114    Also, given that the subspace W is spanned by the…

Q: Q2. Check if the following points lies on the subspace U spanned by the vectors = [2 -1 2]', x2 = [0…

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Q: Find the orthogonal projection of v onto the subspace W spanned by the vectors u;. are orthogonal.)…

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