e: Let A = (4 1 -3 6 What is the null space of A?

Q: Give the demains of and the corresPonding domainsof E+9€F-9+F-9F/9rand/f For the fallowing 2 Fw =…

A: To find the domain of each function

Q: Example 99 Consider the set M = {(x, y) = R²: x,y #0}. Find M. Solution:

A: The orthogonal complement are subspace

Q: Prove that for any vector x, we have (a) x₂ ≤x≤√₂x₂. 1 (b) x ≤x≤x n

A:

Q: Linear operator T: R2 R2 is defined by T(1,0) (1,4), T(1,1)-(2,2) T one-to-one?

A:

Q: (1 1 2 2) Let A = 1 2 3 1,v1 = 1 3 4 0, v2 = Which amongst vı, V2, V3 is an element of null(A)? a)…

A:

Q: i: Let X and Y be topological spaces. Show that X Y X -Y but X -Y X - Y.

A:

Q: Elaborate on the role of duality in topological spaces.

A: Introduction:Topology is a branch of mathematics that studies the properties of spaces and the…

Q: what is reflexive Banach space ?

A: Banach space: A normed linear space N,· is called banach space if it is complete metric space. A…

Q: Prave R is banach Space with explanation.

A: Banach space: A complete normed linear space is called Banach space. Given:  To show:  is Banach…

Q: 7 - 32 0 - 36 ? -4 1 Is - 4| in the null space of A= 9 2 3 - 4 Is it in Column space of A?

A: Suppose A is any mxn matrix then Null(A) is the collection of all vectors y of order nx1 such that…

Q: Prove that the set A = {x ∈ R | 3x + 2 ≤ 0} has a maximum

A:

Q: Find the locus of the points z that satisfies 3(22 + z?) + (8 – 16i)z + (8+ 16i)z – 10|z|? = 16.

A:

Q: The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal…

A:

Q: Let A = {AC R| A is countable} and Az = {A c R| A° is countable} which is a measurable space? (R,…

A:

Q: Let A = 1 -X -X X 1 -x X X -1 Find x such that A is invertible.

A:

Q: the quotient space of a T0−space necessarily T0

A:

Q: Let u, ER" prove that ||ū + ï|| ≤ ||ū|| + ||v||

A:

Q: Determine the null space of A and verify the Rank-Nullity Theorem. 1 -4 9 9 -7 A = -1 2 -4 1 5 -6 10…

A:

Q: 1. Let Q(v2) = {a+b/2| a, b E Q}. Prove that Q(/2) = Q[x]/(x² – 2).

A: We have to show that isomorphism means there is function ϕ such that we have to show that it is an…

Q: Let A, B s R (R represents the real numbers) and A, B both non-empty. Define A + B = {a+b: aEA and…

A:

Q: Find the dimensions of the following linear spaces. (a) RSX3 (b) P6 (c) The real linear space C

A: In general Rm×n is a vector space of dimension m×n. So, Dimension of R5×3 is 5×3 = 15 (a) 15

Q: What is the dimension of the following linear space? 2 x 3 The space R²X

A:

Q: Let U {1,2,..., 100}, and let A consist of 15 unknown numbers taken ran- %3D domly from U. Let B…

A:

Q: Exercise 3.3.16. Prove or find a counterexample to the following statement. Let A, B and C be sets.…

A: Solution of the question is given below.

Q: Given the following fuzzy relations 0.2 0.4 0.6 R₁: 0.1 0.9 0.0 0 0.4 0.8 Compute the following 1.…

A:

Q: 2. Find the (infinity norm) condition number of 2.01 1 (b) A = 3 6.

A:

Q: Let v be the set o all pairs of positive numbers (X,y) With two operations + and C denned as: (x1,…

A:

Q: Determine if the set S = {1+ – 2x2, 2 + 5x- a², a + a2 }is linearly independent or linearly…

A:

Q: The usuall metric space (Rd)isComplete ***

A:

Q: Let f(x) = + 3x? + 4x. a) What is the kernel of f? b) Compute the minimum and the maximum of f.

A:

Q: 1 Assume that Pr[A]=0.45 Pr[B]=0.65, and Pr[A∩B]=0.25  A. Pr[B Compliment ∩ A]= B. Pr[A Compliment…

A: Given, Pr[A]=0.45 Pr[B]=0.65 Pr[A∩B]=0.25

Q: [1030] Let A = 0 120 %3D V be the null space of A, and W= V.What is the dimension of W? 0 0 00 1 3.…

A:

Q: 4 -1 33. The dimension of the nul-space of A = 3 3 is 2 1 1 2 3 (A) 1 (B) 2 (C) 3 (D) 0 (E) 4 142 13

A: Given matrix A=5214−14332211123 We have to find the dimension of null space. For dimension of null…

Q: . Let a := V1+ v3. Find the minimal polynomial of a over Q.

A:

Q: 4 -1 4 33. The dimension of the null-space of A = 3 2 is 2 1 1 1 2 3 (A) 1 (B) 2 (C) 3 (D) 0 (E) 4…

A:

Q: Let X be a pre-Hilbert space then [(x, y)|

A:

Q: Transiate the graphical representation t0 an incquality using the vaviable x. 10 12 14 4- + 30 40 45…

A: Inequality for the first graphical representation is  x1 =14