35. Let V and W be vector spaces, and let T : V → W be a linear transformation. Given a subspace U of V , let T(U) denote the set of all images of the form T(x), where x is in U. Show that T(U) is a subspace of W.

Q: let W be vector that space and suppose USV SW are subspaces of W. Show that W N/u W ~ V/₁ defined…

A: Let W be a vector space over F   and suppose that U⊆V⊆W are subspaces of W. To show: WUVU≈ WV

Q: 7. Let T : V space W. Prove that the range of T is a subspace of W. → W be a linear transformation…

A:

Q: 1. If T is a normal operator on a finite dimensional inner product space V and if W is a subspace of…

A:

Q: Let T: V- W be a linear transformation. now that T(V), the image space, is a subspace of W and now…

A:

Q: 1. Let T: V → W be a linear transformation. Show that T(V), the image space, is a subspace of W and…

A: Let T:V→W be a linear transformation. Prove that the image of T, T(V) is a subspace of W and the…

Q: Let T : V → W be a linear transformation from a vector space V into a vector space W. Prove that the…

A: As per our guidelines we are supposed to answer only first question. Please repost other question…

Q: (b) Let f be a bounded linear functional on a subspace M of a normed space X and let X, E X, x, 4 M.…

A:

Q: If I is a normal operator on a finite dimensional inner product space V and if W is a subspace of V…

A:

Q: Prove that the restriction of a linear operator T to a T-invariant subspace is a linear operator on…

A:

Q: 13. Prove that if V and W are vector spaces and T: V → W is a linear transformation then Range(T) is…

A: Subspace of a vector space

Q: Assume V and W are finite-dimensional vector spaces and Tis a linear transformation from V to W, T:…

A: It is given that, assume V and W are finite-dimensional vector spaces and T is a linear…

Q: Let V, W be finite dimensional vector spaces and T: V → W be a linear transformation. Let A = [T]½…

A: By theorem,  Let u1,u2,.......un be a vector space U. Let V be any vector space. A linear…

Q: Show that if Z subspace of a space Y and Y is a subspace of a space X then Z is a subspace of a…

A:

Q: Let W be a subspace spanned by the u's, an

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

Q: Let V be a real vector space, EC V a subspace, and T : V → a linear transformation satisfying T(E) C…

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 V and W be vector spaces, and let T :V → W be a linear transformation. Given a subspace U of V,…

A: According to the given information, Let T : V->W be a linear transformation. U be a subspace of V…

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: Let u, = 2 |, u2 and uz = 0 Note that u, and uz are orthogonal but that uz is not orthogonal to u,…

A: The orthogonal vector is given as follows :

Q: Let V be finite dimensional vector space and let W c V be subspace with complimentary subspace W'.…

A:

Q: 5. Consider the subspace of M29 defined by V = { : a, b eR}, and the function T : V →→ Pq defin by T…

A:

Q: Suppose H is a n-dimensional subspace. Let A = Let T H H be a linear transformation. Then [T] =…

A:

Q: Theorem 7.35. Let X and Y be topological spaces. For every y E Y, the subspace X × {y} of X × Y is…

A:

Q: Suppose V and W are vector spaces over a field F. If L : V → W is a linear transformation, then show…

A:

Q: Let y : V → W be a linear transformation and let U be a subspace of V. Show that y(U) = {w e W | w =…

A: Given :           φ : V → W is a linear transformation .           U be a subspace of V.…

Q: If Sz is a subspace of R* of dimension 3, then there cannot exist a subspace S2 of R* such that S, c…

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

Q: If V is an n-dimensional vector space, W is an m-dimensional space, and T: V → W is an onto linear…

A: False.

Q: 17. Prove that a subset W of a vector space V is a subspace of V if and only if W + Ø, and, whenever…

A: We Know that Let V be a vector space over the field F and let W E V . Then W will be a subspace of V…

Q: Let V and W be vector spaces over a field F, and let T : V → W be a linear transformation. Suppose…

A: We will use Sylvester's law for determining  the possible pairs. Sylvester's law states that if V…

Q: 1) Suppose that V is a finite dimensional vector space over R. Prove that every T E L(V) has an…

A:

Q: Let V be the subspace of C[a, b] spanned by1, ex, e−x, and let D be the differentiation operatoron…

A:

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

A: Vector space.

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

A:

Q: 2. Decide if the following statement is True or False: Let V be a vector space, and let T: V R be a…

A: The two vectors x⇀and y⇀ are linearly dependent if and only if x⇀=ky⇀ for some k∈R Consider…

Q: Let T:V → W be a linear transformation from a vector space V into a vector space W. Prove that the…

A:

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: Prove that a subset W of a vector space V is a subspace of V if and only if 0 EW and ax+ y E W…

A: First proving the first part: If W is the subset of V and also the subspace of V Claim  : 0 W and…

Q: | Let W be the subspace of R spanned by the vectors 1 11 -2 and Find the matrix A of the orthogonal…

A: The solution is given as

Q: Let T : V → V be a linear transformation. Carefully prove that the image of T is a subspace of V.

A: Since you have asked multiple question, we will solve the first question for you. If youwant any…

Q: Let V be a subspace of M22 with basis 2 0 0] By 3 and let W be a vector space with basis Bw (х, у,…

A:

Q: Please solve this question. Thank you

A:

Q: (a) Prove that the image T(V) = {u : u=T(x) for some x € V } of T is a subspace of W. |(b) If the…

A:

Q: Let uj,..., u, be an orthogonal basis for a subspace W of R", and let T : R" → R" be defined by T(x)…

A:

Q: Let T be a linear transformation from V to W. Prove that the imageof V under T is a subspace of W.

A: Here given T be a linear transformation from vector space V to vector space W over the same field F…

Q: let W={(x,,x,,1):x,,x, are real numbers} is W a subspace of R

A: False

Q: Let L : V → W be a linear transformation from a vector space V into a vector space W. The image of a…

A: By the definition of a vector subspace we need to show 3 properties. 1)0∈L(V1) 2)L(V1) is closed…

Q: (.. Suppose that T :V V is a linear transformation and V is a finite dimensional vector space.…

A:

Q: Suppose V is finite-dimensional. Prove that every linear map on a subspace of V can be extended to a…

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.