Are the following statements true or false? ? For all vectors u, v E R", we have u·v = -v •u. If x is not in a subspace W, then x – projw (x) is zero. For any scalar c and any vector v € R", ||cv|| = c||v||. ? If y = z1 + z2, where z1 is in a subspace W and z2 is in W+, then z1 must be the orthogonal projection of y onto W.

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

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

Q: Is W a subspace of the vector space V? Justify your answer. 1. Wis subset of V = R3 where a W = {…

A: I just used a test of subspace to check the W is subspace

Q: Let H be the set of points inside and on the unit circle in the xy-plane. That is, let {{]***a :x² +…

A: According to the given information, H be the set of points inside and on the unit circle in the x-y…

Q: Let H be the set of points in the second quadrant in the plane V=R^2. That is, H = { (x,y) | x ≤ 0,…

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Q: Let U and W be the subspace of V, with V being the vector space on R body. UnW = {v : v € U ve v e…

A: handwriting solution

Q: b) Suppose that V is finite dimensional and that T E L(V, W). Prove that there exists a subspace U…

A: In a finite-dimensional Vector space, every subspace is finite-dimensional.

Q: Let H be the set of all points in the second and fourth quadrants in the plane V=R^2. That is, H={…

A: ANSWER :

Q: Is W a subspace of the vector space V? Justify your answer. W is the set of all vectors = (3) V =…

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Q: Let V be the set of vectors (x, y, z) E R° such that r(y² + z2) = 0. Is V a subspace of R³?

A: 16. Let V be the set of vectors x, y, z∈ℝ3 such that  xy2+z2=0 We have to check whether V is a…

Q: et X = {(x,y, z): Vx² + y² + z² <1} be a subspace R° with the standard topology and 0 = (0,0,0).…

A: Given,               X=x, y, z:x2+y2+z2 ≤1  be a subspace of R3 with the standard topology and o=0,…

Q: Is W a subspace of the vector space V? Justify your answer. 3. W is the set of all vectors in V= R³…

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Q: Show that S = {(x, Y, z)|4x – y + 5z = 0, x, y, z E R} is a subspace of R³.

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Q: Let S be the subspace of Rn spanned by the vectors x1, x2, . . . , xk. Show that y ∈ S⊥ if and only…

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Q: Q.4 Show that U {(x, -x) |x in R} is a subspace of R. %3D

A: To Prove U is a subspace of vector space V 1) 0∈U  If x, y∈U  and a∈V  2) Closed under vector…

Q: 16. Let V be the set of vectors (x, y, z) E R³ such that r(y? + z²) = 0. Is V a subspace of R?

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Q: 2. Show that U = {(x,y, x) : , y E R} is a subspace of R$. Then, find a linear complement of U in R.…

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Q: Let H be the set of all points in the second quadrant in the plane V=R^2. That is,…

A: Let H be the set of all points in the second quadrant in the plane V=R^2. That is,…

Q: 16. Let V be the set of vectors (x, y, z) E R³ such that r(y?+ z²) = 0. Is V a subspace of R³?

A: Test for subspace for V to be a subspace  If u and v are elements of v then u +av must be in V where…

Q: .a. Is the set of vectors W = {(x,y,z) x,y,z € R,x=z+2} subspace of R³?

A: Given that a) W=x,y,z|x,y,z∈ℝ,x=z+2 of ℝ3. b) W=pt-tq of M22 Check the space W is the subspace of…

Q: The set R' of all column vectors of length three, with real entries, is a vector space. Determine…

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Q: snip

A: We have vector space of all real numbers over real numbers And S is subspace of vector space given…

Q: 12. Let K C R" and let V = { E R" : a1y Vy E K}. Prove that V is a subspace of R".

A: According to our guidelines, we can answer only the first question, the rest can be reposted.

Q: 4) Let X, Y, Z be subspaces of a vector space V and assume that Y CX. Prove that Xn(Y+Z)% =Y+(XnZ).…

A: 4. Given The modular law of sub-space: Let X,Y,Z be subspaces of a vector space V and assume that…

Q: Which of the following sets is not a vector subspace of p? OA. W = ((x,y)x = 2y} O B. W - ((x,y)y-…

A: Subspace of a vector space

Q: Let  S=span(e1),  T=span(e2) and  W=span(e1+e3) be subspaces of R3.  S is orthogonal to T, T is…

A: Orthogonal Subspace

Q: Let W = (a + bx + cx? + dx'la + 2c 0 and b -d 0} be a subspace of P, %3D Then dimension of W is…

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Q: 3) Explain why S, = {(x, y) E R²:y > 0} and S1 = {(x, x²): x E R} are not subspaces of R?. %3D

A: We will solve both these parts using only basic knowledge of subspaces and vector spaces (in clear…

Q: Q.4 Show that U {(x, -x) |x in R} is a subspace of R?. %3D

A: Consider the given set. U=x,-x|xinR Let, consider that, f,g∈V And f, and g is defined as.…

Q: Let W = {(x, y, 1) | x, y, E R}. Is W a subspace of R3?

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Q: 4. Consider C" as a real vector space. It has a subspace U := C x {0}"-1. What is the dimension of…

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Q: 12. Let K C R" and let V = {x € R" : x ly Vy E K}. Prove that V is a subspace of R".

A: " As per the company guidelines we are supposed to solve first question only. Kindly post the other…

Q: snip

A: We can represent the given set again as: W=xyz:x=3y+1 and z=-2y Let  w1=x1y1z1∈WThen,x1=3y1+1 and…

Q: Consider the set ? = ((x, x+ y, 2y)|x,y are real) of vectors in r3 . Is W a subspace? If yes, prove…

A: We have Set S = {(x, x + y, 2y) | x, y are real}. Let u, v be any two element of S then,

Q: Verify that U1 = {(x, y) E R² : y = -2x} is a real vector subspace of R?. hat U2 = {(x,y, z) E R³ :…

A: This a question of linear algebra and vector space.

Q: If v is in R°, then the set of vectors a with v x a = 0 is a subspace.

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Q: - Let V = R and W = {(a, b, c) E V \a + b = c}. Is W a subspace of V? If so, what is its dimension?

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Q: Let W = { |xy <0}. Is W a subspace of R?? If so, use generic terms to demonstrate the necessary…

A: W is not a subspace of R2.

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

A: Solution:Given :     V=ℝ2and      H= (x,y)/ 2x+3y=6   ⊂ ℝ2

Q: W e R"|2, > 0 over the field of reals is a subspace of the specified vector space ER"

A: Given that W=x1x2x3⋮xm∈Rm|x1≥0 and V=x1x2x3⋮xm∈Rm The objective is to show that W is subspace of V.…

Q: Let W be the subspace spanned by u, and u,. Write y as the sum of a vector in W and a vector…

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Q: 4) Suppose u, v E V, where V is an inner product space, and ||u|| = |||| = 1 and (u, v) = 1. Find u…

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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 V = R and W = {x|x = (x1, x2, 1)}, Is W a subspace of V?

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

A: Here we will find out dimension of W.

Q: i) Let V = M2x2(R) and %3D W; = {(: :) ev:a.hceR} and W2 = {(". :) ev :a,be R Prove that W1 and W2…

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Q: If V is a 13 dimensional subspace of R20 then the dimension of V- must be

A: Definition of Orthogonal Complements Let V be a subspace of ℝn. Its orthogonal complement is the…

Q: Let W be the subspace spanned by u, and u,. Write y as the sum of a vector in W and a vector…

A: There are two vectors u1 and u2 is given. write the vector y as the sum of a vector in W and a…