(b) Prove that if V is finite dimensional, then every subspace of V is also finite dimensional. (Hint : reductio af absurdum)

Q: iii) True or False: Every subspace of a complete metric space is complete.

A:

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

A:

Q: Prove that every line through the origin in R3 is a subspace of R3.

A: To prove that every line through the origin in R3 is a subspace of R3.

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

A: **As per our guidelines we are allowed to answer only one question at a time. Kindly please re-post…

Q: If y is in a subspace W, then y - ŷ is not zero. Select one: O True False

A: Solve the following

Q: 2. Let U and W be subspaces of a vector space V such that UUW is a subspace of V. Prove that UCW or…

A: Given that U and W are subspaces of a vector space V such that union of U and W is a subspace of V.…

Q: Show that [0,∞) and (0,∞) as subspaces of R with the usual topology are not homeomorphic.

A: Given: [0, ∞) and (0, ∞) are subspaces of ℝ. To show: [0, ∞) and (0, ∞)are not homeomorphic.

Q: Let H be a subspace of real space R. In terms of a geometrical definition of H, which of the…

A: Given question :- Let H be a subspace of real space R3. In terms of geometrical definition of H,…

Q: If X is the subspace of `∞ consisting of all sequences of zeros and ones, what is the induced metric…

A: We have given that ,  X is the subspace of l∞ consisting of all sequences of zeros and ones. We need…

Q: Show if W is a subspace of the vector space X, then W is convex.

A: Definition used: A set K in a vector space V is called convex if and only if  x, y belongs to K,…

Q: can a subspace of R^n have a dimension less than n.

A: Yes , it can be .

Q: True or False: If U and W are subspaces of a finite dimensional vector space and UnW = {0}, then U°…

A:

Q: 6. Show that the closure Ỹ of a subspace Y of a normed space X is again a vector subspace.

A:

Q: Prove that the set of vectors of the form (a, b, c) where b = a +c is a subspace of R³.

A: According to the given information, it is required to prove that:

Q: Determine if the set W of vectors (X1 , X2) in R such that x1² + x2 = 0 is a subspace of R?.

A: We have to determine whether W is a subspace of R^2 or not.

Q: Prove that the set ? = {(a + b, a -b): a and b are real numbers} is a subspace of ℝ2 with the…

A:

Q: 6. Prove that an open interval (a, b) considered as a subspace of the real line is homeomorphic to…

A: An open interval a,b considered as a subspace of the real line is homeomorphic to the real line.

Q: Let-{ », :x, y, a, b e R and x - a = y – b}.Is W a subspace?

A: We will use the one step test for vector subspace to check if W is a subspace or not. We know that…

Q: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.…

A:

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

A:

Q: 3 show that if w is a subspace of fini a dimensional vector space y and dim (W) = dim (V) W = V.

A: This is theorem in vector spaces

Q: 2. Provide an example of a subset U of V that is a subspace of V, as well as an example of subset W…

A:

Q: snip

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

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: Theorem 6.14. For any a < b in R, the subspace [a, b] is compact.

A: The objective is to prove that any a≤b in ℝ the subspace a,b is compact. Let's prove the theorem by…

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

A:

Q: If y is in a subspace W, then y - ŷ is not zero. Select one: O True O False

A:

Q: Give an example of a vector space V and non-trivial subspaces X.Y. Z of V such thatV =X@Y= X@Z but Y…

A: Give an example of a vector space V and non-trivial subspaces X, Y, Z of V such that…

Q: 4. Prove or disprove that the set of unit 2 vectors of R2 is a subspace of R2under usual operations

A: solve the following

Q: W is not a subspace of the vector space. Verify this by giving a specific example that violates the…

A: The objective is to prove that w is not a subspace of the given vector space.

Q: Let H be an n-dimensional subspace of an n-dimensional vector space V . Explain why H = V

A: We are given that dimension of H is n. And the dimension of V is n. In other words, H is containing…

Q: snip

A:

Q: Suppose V is finite-dimensional, with dim V = n > 1. Prove that there exist 1-dimensional subspaces…

A: Let us consider that (v1, v2, ..., vn) be a basis for V, and let Ui= span(vi), for i ∈ {1, 2, ...,…

Q: Provide a situation in where it shows a union of two subspaces of a vector space V and is not…

A:

Q: 3. Is the set of all vectors (x, y) in R2 with the usual addition and scalar multiplication, a…

A:

Q: Prove that X is completely regular space if and only if it is homeomorphic to a subspace of a…

A:

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…

A:

Q: Prove that a closed linear subspace of a reflexive space is reflexive.

A: This is a problem of functional analysis.

Q: If X is an inner product space, and A is subspace of X then A = A+ True False

A: Let A be a subspace of an inner product space X. Then (i) A⊥ is a subspace of X ; (ii) A∩ A⊥ = (0);…

Q: If M is linear subspace of hilbert space H then M is clos

A:

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?

A:

Q: Show that ifY is a subspace of X, and A is a subset of Y, then the topology a subspace of Y is the…

A: First, we have to show that the topology of A as a subspace of Y is also a subspace of X.

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: Prove that if two linear subspaces are orthogonal, their intersection consists of only the zero…

A: Let W1 and W2 be linear subspaces of a vector space V that are orthogonal to each other. We need to…

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

A:

Q: 5. Prove that for any subspace W of R": dim(W) Hint: the backward direction is obvious. For the…

A:

Q: Let H be a subspace of real space R. In terms of a geometrical definition of H, which of the…

A: Given:

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.