Consider the following subset S={ [x,y,z] ∈ R3 : x=2y, 2y=−z }  of R3 .    Is the zero vector an element of S ? Is S  closed under addition? Is S  closed under scalar multiplication? Is S a subspace of R3?     Answers are yes or no.

Q: 3. Show that the set W = {(x, 0) : x e R} is a subspace of R?.

A: In order to show W is a subspace of R2 , we need to prove W is closed under addition. For any two…

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 =…

A:

Q: Show that the set W : {(x, y, 5x-4y)| x and y are real numbers} is a subspace of R³.

A: We are give, W=x,y,5x-4y| x and y are real numbers To show: W is a subspace of R3 This can be done…

Q: Let S = { (x, y, z, w) E R*; x – 2y + w = 0 and 2x + z – w = 0 } . - Prove that S is a subspace by…

A:

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: 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 W = {a + bx + cx? + dx³| a + b = 0 and c – 3d = 0} be a subspace of P3. Then the dimension of W…

A:

Q: Let W be the set a of all vectors of the form | 0 [2 Is W a subspace of R³

A:

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

A:

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. Determine whether the following are subspaces of the given vector space V. a. S= {ax? + bx + c a+…

A:

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

A: Linear combination of vectors we use to find.

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: For any fixed vector v in R^n, the set V=?x∈R^n :v^(T)x=0 is a subspace of R . Assuming v does not…

A: According to the given information, Let V be a fixed vector in Rn, the set V is given as:

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

A: Given the set of vectors of the form aba+2b , which can also be expressed as:             W =…

Q: If S is closed Subspace of a normed Space (M, f), then prove that the function fo : R defined by f.…

A: If S is a closed subspace of a normed linear space M, f. Prove that the function fQ:MS→ℝ is defined…

Q: How many of the following subsets of the space M2(R) are subspaces of this space? a){A € M2(R):…

A:

Q: Consider the set Y = [-1, 1] as a subspace of R. Which of the following sets are open in Y? Which…

A:

Q: Consider the follouing subset of R4 W = #1 + *3 = 19 +I4 Is W a subspace of R? If it is, describe it…

A: w=x1x2x3x4:x1+x3=x2+x4,x1+x2=x3+x4w is subspace of R4 iff α and β belongs to w then  aα+β also…

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

A:

Q: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

A: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

Q: Is the subset V {(x, y, z, w), y = 3z, w = 2x + 1} a subspace of R*? Show the details of %3D your…

A:

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

A: This staement is true.

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

A:

Q: Suppose V is finite-dimensional and U and W are subspaces of V with W^0 ⊂ U^0. Prove that U ⊂ W.

A:

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

A:

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: Is the following questions below a NEVER TRUE, SOMETIMES TRUE, or ALWAYS TRUE. a.) For 0<m<n, R< R¹.…

A: (a) For 0 &lt; m &lt; n, Rm ≤ Rn.  Justification :- Let x ∈ Rm then x is a m - tuple and Let y ∈ Rn…

Q: Consider the set S = {(x, y, z) ∈ R3 | x - 2y = z}. (a) Show that S contains the zero vector. (b) Is…

A:

Q: 16. Let V be the set of vectors (x, y, z) E R³ such that r(y²+ z²) = 0. Is V a subspace of R³?

A:

Q: Consider the following subset of R": W = {(x1,x2, 83, L4, 25) E R° | r1 = x2 = 13, 14 + 15 = 0}. (a)…

A:

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

A:

Q: Determine the following is the subspace in R’? W ={(x,y,z):x+y+z=0}

A:

Q: 3. Prove that the set S is a subspace of the vector space V V = C(R) and S is the set of f in V such…

A:

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

A:

Q: Show that the given set V is closed under addition and multiplication by scalars and is therefore a…

A:

Q: Let W1 and W2 be subspaces of a vector space V having ilimensions mand n, respectively, where m…

A: Let W1  and W2 be the subspaces of finite dimensional vector space V which satisfy dimW1=m  and…

Q: Are the following statements true or false? ? For all vectors u, v E R", we have u·v = -v •u. If x…

A:

Q: Consider the set S = y ER : 3x – 5y = 0, x + z =- . Is this a subspace of R ? Explain why or why…

A: Given set is S=xyz∈R3:3x−5y=0, x+z=4. We have to check whether the set S is the subspace of R3 or…

Q: Determine whether the given set S is a subspace of the vector space V. v 1. V = Rnxn¸and S is the…

A: We need to determine if the given subsets of the vector spaces are subspace or not. This can be done…

Q: () If S is a subspace in R that is spanned by then S is spanned by: O a. 1 and 0. 1 1 and 1 c. and…

A:

Q: Prove that W1={ (al, a2, ..., an)e Fn: al+a2+ +an=0} is a subspace of Fn, but W2= { (al, a2, ...,…

A:

Q: 7. Consider the subset S of R³, S = {(x, y, z) | 2x – y + 3z = 0}. Show that S is a subspace and…

A:

Q: Let W {a + bx + cx2 + dx³| a+b = 0 and c- 3d = 0} be a subspace of P3. %3D Then the dimension of W…

A:

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: Consider the following subset S = {[x, y, z] E R³ : x = 7y, y = –1 + 2z} of R³ . Is the zero vector…

A:

Q: 10. Let V = M2×2, W,(R) = {; ev : E V : a, b, с€R = {[°. ] ev : « W2 (R) = V : a €R} then prove that…

A:

Q: Which of the following subset is the subspace of the specified vector space? I. T = {(x,0, z,0) |…

A:

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.