I EXAMPLE 7 Let V be a vector space over F and let v, V2, . .. , V, be (not necessarily distinct) elements of V. Then the subset (V, V2» ..., v,) = {a,v, + a,v, + · · · + a,v, \ q, a» - . . , a, E F} is called the subspace of V spanned by v,, v,, ..., v. Any sum of the form a,v, + a,v, + · · V1, V2, . .. , Vn. If (v1, v2, . spans V. n'n + a v, is called a linear combination of v) = V, we say that {v, V2, . .., V„}

Q: Determine whether the set in P_2 is linearly independent.    S={x^2 , +1, 2 - 1, 2}

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Q: 3. Determine whether the given set is a subspace of M2x2 with real entries. {6 ) la = b}.

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Q: Show that a finite union of compact subspaces is compact.

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Q: Each of the given sets below is not a subspace because at least one of the 3 properties of being a…

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Q: One of the following subsets of R² is not subspace. Select one: а. X1 S={ X1=2x2} b. S={ ,X1 =X2} C.…

A: We will check that which subset is not a subspace of R2

Q: Please define the four fundamental subspaces? Please have your answer types.

A: Suppose that A is a m -by- n matrix that maps vectors in R^n to vectors in R^m. The four fundamental…

Q: Let I be the set of irrational numbers with the usual topology (that is, the topology it inherits as…

A: In topology, a component of a topological space is a maximal connected subset of the space. In other…

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

A: Yes , it can be .

Q: The intersection of an arbitary collection of subspaces of a vector space is also a subspace.

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Q: Given U_1, ..., U_m, a collection of subspaces of some vector space V, recall that U_1 + ... + U_m…

A: Here, in the question, there is a collection of subspaces given. We have to give an example of the…

Q: Find a minimal spanning set for the subspace spanned by the set of vectors: S={, , , }

A: The given spanning set is S=1, 0, 0, -1, 4, 4, 0, 0. We have to find the minimal spanning set.

Q: 1 0 Span 1 2 -3 N -2 -2 5 3 -2 2 9 3 I[s -2 4 0

A: The given spanning set in ℝ5is span10-12-3, -2-25-29, 0-2306, -1-2406. We have to find the dimension…

Q: Determine whether the set Wis a subspace of R4 with the standard operations. Justify your answer. a)…

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Q: Determine if all polynomials P = ao + a,x + a,x² + a3x³ , with ao, a1, a2, az integer, is subspace…

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Q: Is it possible to solve it using words like claim as my professor did in his example in other photo?…

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Q: Let X be a nonempty set with subsets, A, B, and C. Prove that A ∩ (B \ C) = (A ∩ B) \ (A ∩ B). I…

A: Given that X is a nonempty set with subsets, A, B, and C. The objective is to prove that…

Q: Determine whether the given set S is a subspace of the vector space V. A. V=P3, and S is the subset…

A: We need to determine that given subsets are subspaces of corresponding vector spaces.We know that a…

Q: Let M3,3 denote the vector space of all 3 × 3 matrices. Let L3,3 denote the set of all lower…

A:

Q: Q6: Explain whether the following sets are not subspaces of R" for the appropriate value of n. a a a…

A: We have to show that given set are vector subspace or not

Q: Consider the subspaces U = span{[4 0 1],[4_1 −4]} W = span { [-5 3 3], [-5 3],[−5 4 −2]}

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Q: et aj = (1,3, –2), a2 = (5,1, 4), and b = (6, –9,3). a) Project b onto the line through aj. b)…

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Q: The intersection of any two subspace of a vector space is a subspace.

A: we have to prove that the intersection of any two subspace of a vector space is a subspace.

Q: Show that a finite union of compact subspaces of a topological spaces X is compact-

A: We have to show that the finite union of two compact subspaces of a topological space X is compact.

Q: Let U_1, ..., U_m be subspaces of some vector space V. Show that U_1 + ... + U_m is the smallest…

A: ●Additive identities:0∈ U1 ,......,0∈ Um . ●Closed under addition: u1 ,w1 ∈ U1 ,....,um  ,wm∈ Um.…

Q: 6. Are the following sets subspaces of P₂? Why or why not? (a) U = {p = P₂|p(3)=0} (b) W = {ao + a₁x…

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Q: Find the orthogonal projection of v= [15 8 0] onto the subspace V of R^3 spanned by [-1 -2 -2] and…

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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.