2) Let V=R²³: W= { x = (X₁₁ X ₂₁ X 3) ER ³²: 2x₁ - x₂ + 5*3=0}. Prove that W is a subspace and find complimentary subspace. Is it (complementary subspace) unique?

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: Answer the following: A SIAII OT VOITOUG 4. Determine if the odd functions (f(-x) = -f(x)) are a…

A:

Q: Suppose U, W,, W, are subspaces of a vector space V. Show that (Un W,) + (UnW,)Cun (W, + W,) Find…

A:

Q: Let P2 be the vector space of all polynomials of degree 2 or less, and let H be the subspace spanned…

A:

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…

A:

Q: 3. Determine whether X1 W E R? : x² + x2 = | X2 is a subspace of R².

A:

Q: If v1 and v2 are in R" and H=Gen{v1, v2}, then H is a subspace of R" Justify

A: Given that : and are in  Let Now, we  have to recall  the definition of subspace of a…

Q: Let (X, F) be a space. Prove that a subspace of a subspace is a subspace of (X, F).

A: Here given X, I is a topological space.

Q: B 1 Find the dimension of the subspace V span{v1, v2, V3, v4} in R³ where vị = 2 1 V2 = , V3 = and…

A: We find dimension of the subspace V

Q: Let P3(R) denote the R-vector space of real polynomials of degree 2 or less p=a+bX+cX2, where a, b,…

A: Given the polynomial .Let the subspace of  be .The subspace is the set of all polynomials from  such…

Q: Consider the subspaces U = {(x,x) | x ≤ R} and W = {(0, y) | y ≤ R} of R². Show that U + W is a…

A:

Q: 2. Let P3(R) be the vector space of all polynomials of degree 3 or less (and the zero polynomial).…

A:

Q: If V is a linear subspace of R", and x1 , ×2 , ... , Xn E V, then we must have span {x1, X2, Xn} =V.…

A: To determine that the statement, if V is a linear subspace of ℝn and x1, x2,....,xn∈V, then spanx1,…

Q: 4 1 Let V= { X= (X₁₂X²)X₁9X4) ER" : X₁ = a X₁ +bx++exs} Where as boc, are fixed real number show…

A:

Q: Determine whether W is a subspace of V

A: Determine whether W is a subspace of V

Q: Determine the dimension of the subspace of R consisting of all XeR* such that x1 + 2x, = 0 and X3 –…

A:

Q: Find the closest point to y in the subspace W spanned by v₁ and v₂. y = 5 - 1 1 18 V₁ 1 - 2 - 1 3 V2…

A:

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

A:

Q: True or False. Justify your answer. Let U Span{1+x, x+x²} and W = Span{x²} be subspaces of P2 (R).…

A:

Q: Which of the following sets are subspaces of R³: W1 = {(x1, x2, X3) : x3 = 2x1 – 4x2}; W2 = {(x1,X2,…

A:

Q: Let S be the collection of vectors [] in R² that satisfy the given property. In each case, either…

A: Note: Hi! Thank you for the question, as per the honor code, we are allowed to answer three…

Q: Describe all linear maps ƒ : R² → R² that are projections with image the subspace ((1,2)).

A: f: ℝ2→ℝ2Subspace =1, 2

Q: Show that the set W={(x,y,1)|x and y are real} is not subspace of R3.

A:

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

A:

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

A:

Q: Let W be the subspace of Rª defines as X1 X2 W = {x = ER : : x1 – 3x2 + 4x3 – x4}. %3D x3 X4…

A:

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: If W is a subspace of R" and if u is in both W and w then u must be a zero vector.

A: I am going to solve the given problem by using the orthogonal property of a vector.

Q: c) Determine whether W = {(a, b, c, d) E R*| ad – bc = 0} is a subspace of R*. %3D

A:

Q: X1 - X₂ = 0 = 0 X3 X4 = 0 X2 - 2x3 -

A:

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: 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…

Q: Let Wi and W2 be two two-dimensional subspaces of the linear space R³, that is dim W1 W = W1n W2 is…

A: Given W1 and W2be the two-dimensional subspace of linear space ℝ3. And dim W1=dim W2=2 let  x∈W1∩W2…