If A is 3 × 3 with rank A = 2, show that the dimension of the null space of A is 1.

Q: The set of vectors in R? with y = -3x is a real vector space

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Q: Determine if S , 2x – 5y = 9} is a subset of V where V = R² is a vector space.

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Q: 1. Provide an example of a vector space Mmxn(Q) that is isomorphic to the vector space P19(Q).

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Q: _I. Let H and K be Hilbert spaces and A E B(H, K). Show that A is compact if = and only if A* A is…

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Q: Let S = {A2«2| trace(A) = 0}, i.e. S = Show that -a S is a vector space.

A: The solution is given as

Q: Let V be an inner product space, and suppose that x and y are orthogonal elements of V. Prove that…

A: Given:

Q: Suppose V₁, Vm, and W are vector spaces. Prove that L(V₁ × ... X Vm, W) are isomorphic to L(V₁, W) ×…

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Q: If X is an inner product space and A = {0} , then A- = {0}O AO

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Q: If H is a Hilbert space and A € BL (H) is positive, prove that A+21 is invertible.

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Q: Prove that (a) C(B) C Nul(A), where C(B) denotes the column space of B. (b) rank A + rank B < 5

A: Given that A is a 3×5 matrix and B is a 5×6 matrix. Also AB=0.

Q: 6. Consider a normed vector space S = (R,|- |). Show that, for all r, y e S, |z - y| 2 | – l9|-

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Q: 5. Let V be an inner product space, and T E L(V). Prove that null(T) = (Im(T*)). %3D

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Q: What is the dimension of the vector space S=abc detEM2x3(R): a=b=c=0}

A: Dimension of vector space is number of element in basis

Q: . u. ): Is the set S = {r +x-1,22 - r+3,-a+3z-2}) a base for the Pa(r) vector space which is a…

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Q: Linear algebra

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Q: Suppose V is an inner-product space and u, v e V satisfy ||u|| = ||v|| = 1 and (u, v) = 1. Prove…

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Q: b) Let X be an inner product space and x, y e x.Prove that xly if and only if kx + y = |lk | + |y*k…

A: Given that X be an inner product space and x,y∈X Let, x⊥y. The objective is to prove that…

Q: The dimension of the vector space R' is equal to 8. Select one: O True O False

A: Since the dimension of the vector space Rn is n. Because, Rn has n-tuple elements.

Q: Show that if A is invertible and DA = CA, then D = C.

A: Given that, A is invertible and DA=CA.

Q: 10. Show that V = R² with the standard scalar multiplication, but addition defined by (x, , y.) +…

A: We have to show that the set V with the addition operation x1, y1+x2, y2=x13+x233, y13+y233 and…

Q: Consider the real vector space M2x3(R) with the usual inner product (A, B) = tr (B' A) Given the…

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Q: Suppose that W is a finite-dimensional vector space over F and that T:V→W is linear. Prove that…

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Q: Let X and Y be normed linear spaces, and T E B(X,Y). Prove that 38 > 0 such that 8||x|| < ||T(æ)||,…

A: Consider the given information.

Q: Consider the vector space P, of all polynomials of degree = a, az+b, b2, then the inner-product of…

A: If p(x)=a1+b1x and q(x)=a2+b2xThen&lt;p,q&gt;=a1a2+b1b2

Q: Determine if S is a subset of V where V = R3 is a vector space.

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Q: Consider the vector space P,(1)of the polynomials of degree less than or equal to 2. Then check…

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Q: Either use an appropriate theorem to show that the given set, W, is a vector space, or find a…

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Q: is

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Q: s an inner product space, where (a, 5) = 4 – 5i.

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Q: Every finite dimensional vector space is algebraically reflexive. Prove that.

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Q: Show that p(x)=5x − 3x2 polynomial in P2 vector space can be written as a linear combination of the…

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Q: Let X be a normed space and A € BL(X) be of finite rank. Then σe (A) = σ₁(A) = o(A).

A: 1

Q: If M = {x: Lx = 0}, where L is a continuous linear functional on H, prove that M¹ is a vector space…

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Q: Assume thát B is a basis for an finite-dimensional inner product space V, and let a E V. Prove that…

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Q: Let V be a complex inner product space. Prove that (v, w) (lle + w||? + ||iv + w||? + ||¿²v + w||? +…

A: We need to prove given product space.

Q: use the inner product = p(x)q(x)dx. to find the projection of p(x) q(x) = 5x + 2. = S'; x3 onto

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Q: Suppose V is a 4-dimensional vector space, and that T is a linear operator on V with minimal…

A: Dimension of Ker T

Q: Consider the vector space P, of all polynomials of degree = a, a2+b, b2 , then the inner-product of…

A: We have to find the inner product of the following -  p = 2 + 1.7x q = 4.8 + 3.2x

Q: Determine a scalar s such that A2 x = sx when Ax = r x.

A: Given that Ax=rx. Multiply both sides of the above equation by A and simplify as follows.

Q: Either use an appropriate theorem to show that the given set, W, is a vector space, or find a…

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Q: Find the dimension of the vector space.P4

A: Here the given vector space V= P4 or

Q: If p4 is a vector space formed by polynomials of degree less than 3 And  s = { ax^3 + bx^2 +cx + d…

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Q: Consider the linear space C[0, 1] equipped with the norm ||f || = max |f(t)|. te[0,1] Prove that…

A: Given: The linear space C0,1 equipped with the norm, f=maxt∈0,1ft   To prove: There is no inner…

Q: Q3) Let L = Z* and |x ||= max {]x1 - x3l, x2l, 15x,1} Vx = (x1, X2, X3, X4) E Z*. Is (Z*, ) is a…

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Q: If V1, V2, V3, V4, and W are all vector spaces.  Are L(V1x...xV4, W) and L(V1, W) x...x L(V4, W)…

A: Given that: V1,V2,V3,V4,W=Vector space The objective is to verify the given statement. Using the…

Q: Letf(z) 3z-2i. Find the maximum valne of /(=} on the disk z<.

A: Triangle inequality:z1-z2≤z1+z2 and z1+z2≤z1+z2

Q: In a vector space V, prove that Ov=0 for all ve V.

A: This quedtion from topic vector space of linear algebra.

Q: Let X# and t1, t2 are two topologies on X such that 11CT2. Prove or disprove that if (X,t2) is a…

A: Counter Example, The set of real numbers with usual topology. That is,  ℝ,τS usual topology space.…

Q: Let (X, || · ||) be a normed linear space. Prove that (a) B1(0) = B1(0).

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

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.