Every T3 space is regular space. True False

Q: Show that every metric space is an Hausdorff space.

A:

Q: Let X and Y be two compact spaces, show that X × Y is compact

A: It suffices if we show there is a collection of open sets that covers X×Y. Let U be an open cover of…

Q: Show that the space (P is complete.

A: The given problem is to show that the space lp is complete. we can use properties of lp spaces to…

Q: (a) R4x7 (b) The real linear space (c) The space of all diagonal 6 x 6 matrices C² 2

A: Here, we have to find the dimensions of given linear spaces.

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: Let B={(1,-2,1), (4,-7,5), (5,-8,8)], and x=(-6,10,-7) Find [x]B

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Q: a) range 1 0 0 11 00 10 b) range 1 1 01 1 c) span ((1,-1, 1), (−1, 1, 1), (1, 1, −1), (1, 1, 1)) d)…

A: Note : '' As per guidelines we will solve first three subparts. If you want any specific subpart to…

Q: For the given matrix A What is the rref of A. What is the rref of AT A = [1 2 1 1 -1 21 0302 5 1 4 5…

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Q: 3. is Show R with explamtion. Banach space, a

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

Q: A completely regular space is also T1-space 1 True 2 False

A:

Q: Define a compact space.

A: The objective is to give the definition of a compact space.

Q: Prove numbers that the set is aB anach all real брасе.

A: We have to show that the set of all real numbers is a Banach space. Definition :…

Q: 1 -1 -1 1 -1] Let A -1 -1 1 1 1 Find a non-zero vector x in the null space of A. -1 -1 -1 -1 X =

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Q: what is a proper space

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Q: - theorm; – Every inner product space is a norm linear space

A: We have to prove that every inner product space is a norm linear space.

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

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Q: Give example T2 space but not T3 Space

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Q: Find three different bases for the 3-dimensional real space (ususally denoted R3).

A: Basis: Basis of a vector space (V) over a field F (R or C) is linearly independent subset of vector…

Q: Let A = {a₁, a2,..., am} be a CRS modulo m. Show that for any integer r, the set S = {a₁ +r, a2…

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Q: 4={(-21.r):reR} II) B=(2r.-31): te R} III) C={(2r+3,-41) re R} I)

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Q: Can you show an example of Null space?

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Q: Give an example of 3 vectors spaces that are not Rn. Explicitly state the definition of addition and…

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Q: Give an example of a norm space.

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Q: If A and B are distinct points in 3-space then |AB| = - Select one: True False

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Q: ACX A + Ø and A + X. Let B = AU fr(A). .show that for the topology T = {X,Ø, A, B} it happens that…

A: Given A⊂X where X is non-empty and A≠X Suppose, T=X,∅,A,B be a topology.  Here B=A∪frA To prove for…

Q: Show R³ is a banach space.

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

Q: Brove R is banach Space. with explanation.

A: Banach space: A complete norm linear space is called Banach space. Given: ℝ3=x(1), x(2),x(3):…

Q: (5) Let Pe B(H) be a nonzero idempotent. Show that the follow- ing are equivalent. (a) P is a…

A: It is provided that P∈BH is a nonzero idempotent.  We have to the following are equivalent: (a) P is…

Q: Find isomorphic spaces O R and R² O P3 and M2x1 O P3 and M2×3 O P3 and R2 O R' and M2×2

A: We have to choose whether the given spaces are isomorphic or not.

Q: Determine whether the following sets form sub- spaces of R³: (a) {(x1,x2, X3)" | x1 +x3 = 1} (b)…

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Q: the euclidean space R³ is a separable metric space. true or false?

A: A subset M of a metric space X is said to be dense in X if M=X.X is said to be separable if it has a…

Q: Show that if X and Y Hausdorff spaces, then so is the product space X × Y. are

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Q: Every T2 (Hausdor ff) space is T1 space. True O False O

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Q: Show that the closure of a connected space is connected.

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Q: ive example to show that a quotient space of a second countable space need not be second countabl

A: This is a problem of topology.

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.