Prove that if a 1-dimensional subspace W of Rn contains a nonzero vector with all nonnegative entries, then W contains a unique probability vector.

Q: Let S be the set of symmetric 3 × 3 matrices. Determine whether S is a vector subspace of M3×3.

A: The answer is given below:

Q: Let W be the set of all 2 x 2 matrices A = |" 2 such that Az = 0, where z = H. Is W subspace of M22?…

A: Given that W is the set of all 2×2 matrices A such that Az=0 where A=abcd. Here z=11. We have to…

Q: Find a linearly independent set of vectors that spans the same subspace of R' as that spanned by the…

A: Given :           Three vectors that spans of ℝ3 as :                  v1 = -23-2 , v2 = 480 , v3 =…

Q: Let W be the set of all 2 × 2 matrices A = " 2 га b] such that Az = 0, where z = . Is W subspace of…

A: A set W of vector space V over field F is said to be subspace of vector space V if W is itself a…

Q: Let V be the vector space of all (3×3) matrices and W be the set of all (3×3) matrices A for which…

A:

Q: Let X be an n-dimensional random vector and the random vector Y be defined as Y = AX + b, where A…

A: It is widely used in statistics . It has applications in almost all multivariate problems .

Q: Verify that if W is a subspace of an inner product space V and v is in V, then perpw(v) is…

A:

Q: Let W be the set of all 2 × 2 matrices A = " d. such that Az = 0, where z = |. Is W subspace of M22?…

A: Use the properties of a subspace or definition of subspace. Assume that the condition is true and…

Q: Give an example of a subset (not subspace) of R3 that has an infinite number of elements, is closed…

A: A subset W of vector space V over the scalar field K is a subspace of V if and only if following…

Q: Construct the vector space of all 3-tupies over GF(3). Form a two-dimensional subspace and its dual…

A: Given, the vector space of all 3-tuples over GF(3).

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

A: Yes , it can be .

Q: If W is a subspace of V and B is a hasis for V, then some subset of B is a basis for W. Select one:…

A:

Q: Prove that the set of all n x n matrices A such that AT = -A is a subspace of Mn,n.

A: We have to show that set of all skew symmetric matrix form subspace of Mnn

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

A:

Q: The empty subset of any vector space V is not a subspace of V. True False

A: The empty subset of any vector space V is not a subspace of V The given statement is false.

Q: Find a linearly independent set of vectors that spans the same subspace of R as that spanned by the…

A: A collection of vectors spans a set if every vector in the set can be expressed as a linear…

Q: Let W be the subspace of all symmetric matrices in M2,2. What is the dimension of W?

A:

Q: Prove that the set of mxn upper triangular matrices is a subspace of Mmxn(F).

A: We have to prove that -

Q: Let P be the projection matrix corresponding to a subspace S of Rm. Show tha PT = P

A:

Q: Determine whether the subset W of Max2 is a subspace. S is the set of all matrices satisfying an +…

A:

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: Let S be a subset of a vector space V. Show that span (S) is the smallest subspace containing all…

A: Given that, S be a subset of the vector space V. We have to show that spanS is the smallest subspace…

Q: Any nonzero subspace W C R" has an orthonormal basis. True O False

A:

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: Consider the set U2 : a, b, d e R} of 2 x 2 upper triangular matrices with real number entries.…

A:

Q: 9. Find a basis for the orthogonal complement of the subspace W=Span

A:

Q: Let V be the vector space of symmetric 2 × 2 matrices and W be the subspace -4 -3 -3 5 W = span{ |}.…

A:

Q: 5. Let W be a subspace of the vector space V. Prove that the zero vector in V is also the zero…

A:

Q: If V is any three-dimensional subspace of R°, then V has infinitely many bases.

A:

Q: he zero vector in U is a subspace of V.

A: Inroduction a. Zero vector is a subspace of every vector space.

Q: Suppose that A is a fixed 2x2 matrix. Let U = {X ∈ M22 | AX = XA} Question: Prove that U is a…

A:

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: Let Mnn be a vector space of n x n matrices. Which of the following is not a subspace of Mnn?

A: The vector space of nxn matrix is Mnn.

Q: Show that an arbitrary linear combination of two states Vs in the vector space V with the same…

A:

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

A:

Q: If M is a closed subspace of a Hilbert space H then H %3DМӨМ'.Prove.

A: Complete inner product space is called Hilbert space

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: Show that the set of all polynomials of degree at most 3 forms a subspace of Pn.

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: 12) Show that subset W of a vector space V is a subspace of V if and only if span(W) = W.

A:

Q: Prove that the intersection of two subspaces of a vector space is a subspace.

A: The objective is to prove that the intersection of two subspaces of a vector space is a subspace.…

Q: Find a linearly independent collection of vectors which spans the following subspace of R4: 2 3 -3…

A:

Q: Find a basis for the subspace W of R³ orthogonal to {(1,1,3, 4, 1), (1, 2, 1, 2, 1)}.

A: We Use Gram Schmidt Process to Find Orthogonal basis. Use Standard Inner product as <(u1, u2),…

Q: Find a basis for the subspace of R3 spanned by S.S = {(2, 3, −1), (1, 3, −9), (0, 1, 5)}

A:

Q: Find a basis and the dimensi on of the subspace W of V = M22 spanned by A B

A: V=M2X2R SPANNED BY A=1-5-42, B=11-15, C=2-4-57, D=1-7-51 Solution:- To find: A basis and the…

Q: Find a basis for W, the orthogonal complement of W, if W is the subspace spanned by 2

A:

Q: Determine whether the subset of Mn.n is a subspace of Mn.n with the standard operations of matrix…

A:

Q: Let V be the vector space of symmetric 2 x 2 matrices and W be the subspace -4 -3 3 W = span{ -3…

A: There are so many vectors does no belong to W but belong to V.

Q: 2. Determine if the set of all 2 x 2 matrices where tr(A) = 0 is a subspace of the vector space,…

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.