Let W = { |xy <0}. Is W a subspace of R?? If so, use generic terms to demonstrate the necessary properties. If not, give a specific example where one of the necessary properties fails.

Q: What is the dimension of the subspace H of R spanned by the given ved V1, V2 and v3 2 3 Vị = -8 V2 =…

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Q: Show that W = {(x1, x2): x1 ≥ 0 and x2 ≥ 0}, with the standard operations, is not a subspace of R2.

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Q: Show that W is a subspace of R¹.

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Q: Consider the subspace V of R4 given by 1 3 1 V = span || -1 1 1

A: Solution:-

Q: 2. Find the closest point to y in the subspace W spanned by vị and v2. 3 1 -2 y V2 13 3

A: To find  The closest point y in the subspace W spanned by v1 and v2.

Q: (c) Find a subspace U such that R³ = W₁ OU.

A: C. Hint: Use the result of part (b) in the part (c) and obtain the required result.

Q: Let Z E R³ | y Prove that Z is a subspace of R³. = Z

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Q: Let W = {a + bx + cx² + dx°|c - 3d = 0} be a subspace of Pa. Then the dimension of W is equal to 3.…

A: PnF is set of all polynomials of degree nand smaller . PnF denotes the vector space over field F…

Q: If A is a normed subspace of E, proof that bd(A) is a subspace of X.

A: Please check the answer in the next step

Q: a. . Is the set of vectors of the form a subspace of R³? Show your proof. La+2b

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Q: Determine is Question 2 whether W = W = {(a; z { (a; 2; 6) | α; 6ER} subspace of R

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Q: Show that H= {p € P3: p(0)+p(1) = 0} is a subspace of P3. %3D

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Q: What is the dimension of the following subspace U of R2x2 ? 3 U = span({ }) -2 -2

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Q: b) Let W = { 0) * x, y e R}. Show that W is not a subspaces of M2x2. %3D

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Q: let H= { a b 0 \ a ∈ R and b∈ R.} Show that H is a subspce of R3 by showing that it…

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Q: 4. Let A be a real m x n matrix. Show that U = {x E R" : Ax = 0} is a subspace of R".

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Q: If S and T are subspaces of V, then SUT is a subspace. Select one: True False

A: Given S and T are two subspacewe have to check whether S∪T  is subspace or notClearly  S∪T  is…

Q: a Is the set of vectors of the form a subspace of R³? Show your proof. La+2b

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Q: Let W = {a + bx + cx² + dx³| c – 3d = 0 } be a subspace of P3. Then the dimension of W is equal to

A: As given W be a subspace of P3.

Q: Let W = :a +5c = 0} be a subspace of M22. %3D Then dimension of W is equal to: None of the mentioned…

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Q: a. Is the set of vectors of the form a subspace of R3? Show your proof. La+2b|

A: Since you have asked multiple questions, we will solve the First Question for you. If youwant any…

Q: Suppose that X = ER' :z + y + z = 0 and ² + y² = z² Determine (with justification) whether X is a…

A: Given  X=xyz∈ℝ3:x+y+z=0 and x2+y2=z2

Q: It can be shown that the set of all polynomials: p(t) on R is a real vector space. dp Is the set {q|…

A: The objectives to show the given set is a subspace

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: Let W = {a + bx + cx2 + dx'|c-3d = 0} be a subspace of P, Then the dimension of W is equal to %3D 4.…

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Q: If S, is a subspace of R* of dimension 3, then there cannot exist a subspace S, of R* such that Si c…

A: S1, S2 are subspaces of R4 such that S1⊂S2⊂R4 with S1≠ S2 and S2≠R4.

Q: Determine whether the set of all 3-tuples of the form (x, y, z), where z = x – 2y, form a subspace…

A: If W is a subset of a vector space V over a field F, then W is a subspace of V, only if it satisfies…

Q: 2. Is the set of vectors of the form a b a+2b a subspace of R³? Show your proof.

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Q: The set {(x,3): x is in R)} is a subspace of R2

A: The given set is x,3: x is in R. It is known that any subspace should contain identity element.

Q: Let X = c, the space of all conver gent sequenes and M == (xn) such that , = 0 then M is not…

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Q: Suppose that U cr|xyer F4 y EF, U is a subspace of F4- you don't have to! Find a subspace V of F4…

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Q: and T are subspaces of V, then SUT

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Q: Let W be the set of all vectors of the form Na subspace of R³ 0 2

A: Here given that  vector  [a 0 2]    We know that condition of subspace  w is zero vector belong…

Q: II. Decide which of the following are subspaces of corresponding vector spaces: C R?

A: Vector space.

Q: Let H be the set of all vectors of the form 3s . Find a vector v in R3 such that H = Span {v}. Why…

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Q: Given a Vector space V, if S is to be shown to be a subspace of V, is it enough to show that the…

A: Given a Vector space V, if S is to be shown to be a subspace of V, is it enough to show that the…

Q: Given two n xn matrices A and B, prove that the set S = {x € R" : ABx = 0} is a subspace of R".

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Q: Let A be an n x n matrix. Verify that V = {x e R" : Ax = 3x} is a subspace of R".

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Q: Let V = M22 and W = {invertible 2 x 2 matrices}. Determine whether W is a subspace of M2. %3D

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Q: ) Do there exist subspaces S,T, and U of R³ such that S 1 T, SI U, and TiU? if so, provide an…

A: I have considered te subspace generated by orthogonal basis vector.

Q: Which of the following subsets of R is a subspace? {(:) \ z + y° = 0} {(:) \ +2y = 0} {(;) \ z +2y =…

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Q: 1- Show whether the set of polynomials a, +at +at+at ,where a, =a, +a, is a subspace of P3 or not ?

A: Let V be a vector space over the field F. A non-empty subset S of the vector space V is said to be a…

Q: ) The degree of pa is equal to the dimension of the cyclic subspace Z (a ; T) .

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Q: Let W = {a + bx + cx? + dx*Ia + b = 0 and c – 3d = 0 } be a subspace of P3. Then the dimension of W…

A: Use the concept that dimension of subspace is number of free variables.

Q: Let p(x) be a polynomial in P4. Let S be the set of all polynomials from P4 with p(0) = 0. Is Sa…

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Q: Is the set M={1/n | n ε z+} compact as a subspace of R?

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Q: 1. Decide if the following subsets W C V are vector subspaces or not (justify) (a) W = x €RCR?. COs…

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

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Q: The subspace V of R4 is spanned by the vecto (a) Use the Gram-Schmidt process to find a

A: Here, in the question there is given some of the vectors with a subspace spanned by the vectors. We…

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…

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.