4. Let H be the set of 2 x 2 whose trace is equal to zero. That is a11 a12 Н- E M2(R) | a11 + a22 = 0 a21 a22 Prove that H is a subspace of M2(R).

Q: 1. Consider the following subspaces of R³: U = {(x,x,y): x,y ER} and W = {(x,-x, 2x): x, y ≤ R} (a)…

A:

Q: The subset in the form where x and y are real numbers, is a subspace of R2 if Select one: O x + 0…

A:

Q: Show that the set W : {(x, y, 5x-4y)| x and y are real numbers} is a subspace of R³.

A: We are give, W=x,y,5x-4y| x and y are real numbers To show: W is a subspace of R3 This can be done…

Q: 3. Let W = M2x2(C) with zero determinant. Show that W is not a subspace of V = M2x2(C).

A:

Q: Prove that every line through the origin in R3 is a subspace of R3.

A: To prove that every line through the origin in R3 is a subspace of R3.

Q: 1. If T is a normal operator on a finite dimensional inner product space V and if W is a subspace of…

A:

Q: 8. Check the following set of matrices W A = a1 a2 E RMXm|Det(A) = 0 ... over the field of reals is…

A: Subspace of a vector space is a non-empty subset of the vector space which itself a vector space…

Q: Determine, with justification, whether the following sets are subspaces of R4. --| I4 = 0 (a) S (b)…

A:

Q: Consider L = I – A with A : X → X is a compact linear operator in a normed space X. Show that the…

A: The given problem is related with functional analysis. Given that L = I - A , where A : X → X is a…

Q: Determine whether the following are subspaces of C[−1, 1]: The set of continuous nondecreasing…

A: According to the given information,

Q: 4. Assume that T E L(V) and that U1 and U2 are T-invariant subspaces of V. Prove that U1 n U2 is…

A:

Q: 5.7.5 ND Suppose that V has an inner product, and that n is in V. Show that the set Vo of all…

A: We will use the definition to prove that, V0 is a subspace of V, keeping in mind the properties of…

Q: 2.14. Prove that two subspaces V1 and V2 are linearly independent if and only if Vịn V2 = {0}.

A: V1 and V2 are two subspace.

Q: Z. Suppose A and B are m x n matrices. Prove that C(A) and C(B) are orthogonal subspaces of R" if…

A:

Q: 10. Under the usual operations in R2, which of the following subsets is a subspace? (a) y = 2ar…

A:

Q: Determine is Question 2 whether W = W = {(a; z { (a; 2; 6) | α; 6ER} subspace of R

A:

Q: Find the orthogonal projection of v = projv (v) = [B] 9 onto the subspace V of R³ spanned by 4 and…

A:

Q: 1. Determine whether w - {[:]•r:a-4-0} X1 E R? : : x1 – 23 = X2 is a subspace of R?.

A:

Q: 4. Consider the following subspaces of P. H = Span{1+t, 1– t°} and G= Span{1+t+t°, t – t°, 1+t+t}…

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Q: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

A: Prove that W1 = {(a1, a2, . . . , an) ∈ Fn: a1 + a2 + · · · + an= 0} is a subspace of Fn , but W2 =…

Q: 4. Consider the following subspaces of P. H = Span{1+t, 1-t3 } and G = Span{1+t+t 2, t – t 3, 1+t+t…

A:

Q: b) Let W = { 0) * x, y e R}. Show that W is not a subspaces of M2x2. %3D

A:

Q: 3.8 Using the Gram-Schmidt method, turn the basis B = (b1, b2) of a two- dimensional subspace U C R…

A:

Q: 4. Let A be a real m x n matrix. Show that U = {x E R" : Ax = 0} is a subspace of R".

A:

Q: 4. Consider the following subspaces of P. H = Span{1+t, 1– t°} and G=Span{1+t+t°, t – t°, 1+ t+t®}…

A:

Q: Let W = :a +5c = 0} be a subspace of M22. %3D Then dimension of W is equal to: None of the mentioned…

A:

Q: (1) Suppose U and W are both five-dimensional subspaces of R°. Prove that UnW + {0}.

A:

Q: Suppose that X = ER' :z + y + z = 0 and ² + y² = z² Determine (with justification) whether X is a…

A: Given  X=xyz∈ℝ3:x+y+z=0 and x2+y2=z2

Q: If W is a 2 x 2 matrices A with trace equal to zero is a subspace of M2,2. Prove that span X = W.…

A:

Q: Let S1 = {(xy)/ x22y-1}, S2 = {(x,y)/x <3, 1s y 5} and S3 = {(x,y)/ O< x<3, 1sys5}. %3D Then: S1 and…

A: We use the result  compact subset of R is closed and bounded.

Q: 3.12. Show that for a subspace E we have (E-)- = E. Hint: It is easy to see that E is orthogonal to…

A:

Q: Let  S=span(e1),  T=span(e2) and  W=span(e1+e3) be subspaces of R3.  S is orthogonal to T, T is…

A: Orthogonal Subspace

Q: If Sz is a subspace of R* of dimension 3, then there cannot exist a subspace S2 of R* such that S, c…

A: S1, S2 are subspaces of R4 such that S1⊂S2⊂R4 with S1≠S2 and S2≠R4.

Q: Show that the set W={(x,y,1)|x and y are real} is not subspace of R3.

A:

Q: 4. Let H be the set of 2 × 2 whose trace is equal to zero. That is [a11 a12 Н- E M2(R) | a11 + a22 =…

A:

Q: 2 Suppose V is finite-dimensional and dim V > 1. Prove that the set of noninvertible operators on V…

A:

Q: Find a basis and the dimension of the subspace W of P3= {ax3+ bx2 + cx + d} defined by W = {PEP3:…

A:

Q: 2. Determine which of the following is a subspace of R". (a) {(x1,x2,...X) 1 x = 0} (b) {(x1,…

A:

Q: If x is in W, then x is orthogonal to some vector in a subspace W

A: SOLUTION-

Q: Be F1(x, x, x) e R³ and F2(a, b,0) e R³ a)show that F1 and F2 are subspaces of R b)show that R° = F1…

A:

Q: Be F1(x, x, x) e R³ and F2(a, b,0) e R³ a)show that F1 and F2 are subspaces of R3 b)show that R' =…

A: Solution of the problem as follows

Q: Let P, be the set of all polynomial functions in P(R) of the form ·+ anx", for some fixed positive…

A:

Q: Prove that W1={ (al, a2, ..., an)e Fn: al+a2+ +an=0} is a subspace of Fn, but W2= { (al, a2, ...,…

A:

Q: Let S = {[" e Mzxzi a = d and b + c = 0} be a subspace of M2x2- Then the dimension of S is equal to:…

A:

Q: Find the orthogonal projection of -13] 15 onto the subspace W of R4 spanned by 4 and 5 -2 -2…

A:

Q: Suppose V is finite-dimensional and U is a subspace of V such that dim U = dim V. Prove that U = V.

A: A set B is a basis of a vector space V if its elements are linearly independent and every elements…

Q: Let  S=span(e1),  T=span(e2) and  W=span(e1+e3) be subspaces of R3.  S is orthogonal to T, T is…

A: Given that S=spane1 and W=spane1+e3. To prove that S is orthogonal to W. Let x,0,0∈spane1 and…

Q: Let A e Mnxn and W be the set of all solutions for the homogenous linear system AX = 0. Prove or…

A: Given, A∈Mn×n and W be the set of all solutions for the homogenous linear system AX=0. We have to…

Q: Prove that is U1 and U2 are subspaces of V then dim(U1 + U2) dim(U1)+dim(U2)-dim(U1 N U2). Using…

A:

Q: (5) Suppose that T E L(V) is such that all subspaces of V of dimension dim(V) – 1 are T-invariant.…

A: Please check step 2 for the solution.!

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.