What is the projection of v = e₁ +2e2 − 3e3 on H : x₁ + 2x2 + 4x4 = 0? et V be a subspace of R". Then show that dim V = n − 1 if and only if V = {x r some a # 0.

Q: Determine if (x, y, z, t) ∈ R^4 such that y = −x and z = 0, and t = 2x form a subspace of R^4

A: Use the necessary and sufficient condition for a subspace of a vector space

Q: 6. Let Vi and V2 be subspaces of R" such that Vi C V2. Show that any vector v E R" can be uniquely…

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Q: What is the matrix P = (P₁) for the projection of R³ onto the subspace V spanned by the vectors a₁ =…

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Q: Let V be the set of all pairs (x,y) of real numbers together with the following operations: (x1,y1)…

A: Given  be the set of all pairs  of real numbers together with the following operations:

Q: Find the closest point to y in the subspace W spanned by u₁ and ₂. y = 19, Un 28 国 26 20 -2 -26…

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Q: Verify that V5 is a subspace of RS

A: Please check step 2 for solution.!

Q: The vector x is in a subspace H with a basis ß = {b₁,b₂}. Find the ß-coordinate vector of x. b₁ =…

A: The given basis is b1,b2=1-2, -53. To Find: beta coordinate vector of x=19-17 w.r.t the above basis.

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}

Q: 2. Find the projection from R³ onto its subspace spanned by (1, 2, −1)ª and (1,1,2).

A: Given that We  have  to  find  the projection from   onto  its  subspace spanned  by  .Let   be…

Q: b) Is W = {f(x):f is a polynomial with degree <3,f (0) 0} a subspace of P: all 3 ? Explain your…

A: To see if the given subset is a subspace of the vector space of all polynomials of degree less than…

Q: 5. Let H = {x E R³| x1 + x2 + X3 = 0}. Prove that H is a subspace of R3.

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Q: The set of all vectors in R squared  that satisfy 3x+4y=1 is a subspace of R squared . True False

A: 1. The set of all vectors in R squared that satisfy  3x+4y=1  is not a subspace of R squared.  i.e.…

Q: Which of the following is a subspace of p2? O a. R2 x, e R W = Зх, + 1 1. Ob. W = ER| x*2 2 0 C. ER?…

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Q: Show how v is orthogonal to any linear combination formed by w1 and w2

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Q: Find the orthogonal projection of onto the subspace W of R4 spanned by projw() = --B 7 010

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Q: 2t 6. Let H be the set of all vectors Show that H is a subspace of R³. What is the -t dimension of…

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Q: Let U, W be subspaces of R" and assume UnW = {0„}. Prove that dim(U + W) = dim(U)+ dim(W). Here U +…

A: We have to prove given result:

Q: Find the orthogonal projection of onto the subspace W of R4 spanned by projw (7) 0 -8 v= 0 0·00 ||

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Q: Show that every subspace of ℝ is either {0} or ℝ.

A: Given: ℝ. To Show: Every subspace of ℝ is either {0} or ℝ.

Q: Prove that the the solution to the equation 2x – y+z= 0 is a subspace of R³.

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Q: Suppose that X = eR' : 2 + y + z = 0 and r²+y² = z² } . Determine (with justification) whether X is…

A: We have to check whether the set X is a subspace of the vector space ℝ3. We have to give a…

Q: Find a linear mapping G that maps [0, 1] x [0, 1] to the parallelogram in the xy-plane spanned by…

A: and .We have to find .

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

A: Solution of the problem as follows

Q: Find the closest point to y 2 4 0 in the subspace W spanned by ₁ = 2 0 -1 -3 and ₂ = 7 -2 3

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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: Let vi # v2 E R" and define W1 Span{v2}. Use the definition to show W+n W, is a subspace of R".…

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Q: 3) Find a basis for the subspace of R that consists of the intersection of the planes X-3Y = 5Z and…

A: This is a problem of vector space.

Q: Find the orthogonal projection of onto the subspace W of Rª spanned by projw() = v= 4 -2 -13] 3 15…

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Q: 5. Let P2 be the vector space of polynomials of degree at most 2. Determine if the vector v = = 10x…

A: Consider the given vector v is in the given subspace. Consider the real numbers a, b such that the…

Q: Show {s,t,4s-3t)|s and t are in ℝ} is a subspace of ℝ3

A: Let A= {s,t,4s-3t)|s and t are in ℝ}To show A is vector space it is enough to show that   v1,v2…