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: XAB = BAX} is a {X Let A and B be fixed 2 x 2 matrices. Prove that the set W = {X: XAB = subspace of…

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Q: 4. The set W of polynomials of the form a² + α₁x + α₂x² + 3x³ + α4x4, where a₁ = a₂ and a₁ = az is a…

A: 4. Given that the set W is of the polynomials a0+a1 x+a2 x2+a3 x3+a4 x4 where a0=a2 and a1=a3 is a…

Q: *1. Which of the following are subspaces? Justify your answer in each case. (d) (x € R³ : x² + x² +…

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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…

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Q: Suppose W₁, W₂ are subspaces of a finite-dimensional inner product space. Prove the following: (a)…

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Q: (d) The set U = [Select] C : a, b, c, d e R, ab = cd is a subspace of R¹.

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Q: 1. Suppose that A is an nxn non-zero, real matrix and 2 is a fixed real number. Let E ={x=R* : AT =…

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Q: 4. Decide which of the following sets of vectors are subspaces of R2. Justify your answers. a₁ such…

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Q: Let and V=M₂x2(F). W₁ = {(2) €V: a,b,cEF}. 0 { (-ai) € V: ab€ a, W₂ 1 F}. Prove that W₁ and W₂ are…

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Q: Describe the four subspaces of R³ associated with the matrices: [010] 1 1 0 A 0 0 1 (I+A) 0 1 1 001…

A: Matrices are as follows:To Find: The four subspaces of  associated with the matrices.

Q: 10. Prove that W₁ {(a₁, a2,..., an) € F": a₁ + a₂ + + an = 0} is a subspace of Fr, but W₂ = {(a1,…

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Q: 10. Prove that W₁ = {(a₁, a2,..., an) € F": a₁ + a2 + subspace of F", but W₂ is not. + An = 0} is a…

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Q: Find bases for the following subspaces of F5: and W₁ = {(a1, 02, 03, 04, 05) € F5: α₁ - 03 - α4 = 0}…

A: Given subspaces areWe have to find the bases for the above subspaces also we have to determine their…

Q: -F -5 8 generated by v₁ and v2. 3. Let vi 2 3, V₂ = -5 and w = 8 B 2 Determine if w is in the…

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Q: 4. Let S be the subspace of P₂ defined by S = span{1 − x, 1 + x, 5, 1 – x², 2x}. Find dim(S), and…

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Q: 9. Let F be a field. Find bases of the following two subspaces of F5 : W₁ = {(a₁, a2, a3, a4, a5) =…

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Q: 5. Let V = M2×2, {: ]=v W1(R) = E V : a, b, c ER W,(R) = {|% ev : a, beR E V : a, bER then prove…

A: To prove that a set S is a subspace, we show The zero vector is in S Closed under addition…

Q: 4. The set W of polynomials of the form α₁ + α₁x + α₂x² + α3x³ + axª, where a = a₂ and a₁ = az is a…

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Q: 12. Prove that the set of mxn upper triangular matrices is a subspace of Mmxn (F).

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Q: 5. If W is subspace, if A = : a. (A + B) is subspace these [8] € W and B = Non of above [8] € W,…

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Q: 1. a. b. x1 + x2 x1 2x1 - X2 Find the solution set of the following system of linear equations : - -…

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Q: Explain why the subset а —b S : a + b а,b€ R -2b — За of R³ is or is not a subspace of R³.

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Q: Let V₁ = V₁ and v2. 1 -4 , V2' = -2 3 and w= -3 Determine if w is in the subspace of R³ generated by…

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Q: Let S = {E} " +=0}. such that ₁ + X2 + X3 = Either prove that S is a subspace, or find a…

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Q: 5. Find the dimension of the subspace of P, spanned by the given set of vectors: (a) {r2, r? +1, x²…

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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.