Help with question 28 122Chapter 2c. Show that the matrix25. A =Linear Combinations and Linear Independence26. A =cannot be written as a linear combination ofM₁, M2, and M3.In Exercises 25 and 26, for the given matrix Adetermine if the linear system Ax = b has a uniquesolution.M =1 20-1 0 32 1 232 41 -1 40 2 -4for all x implies that0 33 1In Exercises 27-30, determine whether the set ofpolynomials is linearly independent or linearlydependent. A set of polynomialsS = {p₁(x), p2(x), ..., Pn(x)} is linearly independentprovidedC₁p1(x) + C₂ p2(x) + ··· + Cn Pn(x) = 0C1 C₂ == n = 027. p₁(x) = 1 p2(x) = −2+4x²P3(x) = 2x P4(x) = -12x + 8x³28. p₁(x) = 1 p₂(x) = xP3(x) = 5 + 2x - x²29. p₁(x) = 2 p2(x) = x p3 (x) = x²P4(x) = 3x - 130. p₁(x) = x³ - 2x² + 1 p₂(x) = 5xP3(x) = x² − 4 P4(x) = x³ + 2xIn Exercises 31-34, show that the set of functions islinearly independent on the interval [0, 1]. A set offunctions S = {f₁(x), f2(x), . fn (x)} is linearlyindependent on the interval [a, b] providedC₁ f1(x) + c2f2(x) + ... + Cn fn(x) = 0for all x = [a, b] implies thatC1 C₂ == n = 031. f₁(x) = cos лx ƒ₂(x) = sin лx32. f₁(x) = et ƒ₂(x) = e¯xf3(x) = ²x33. f₁(x) = x f₂(x) = x² ƒ3(x) = et34. f₁(x) = x f2(x) = exf3(x) = sin лx35. Verify that two vectors u and v in R" are linearlydependent if and only if one is a scalar multipleof the other.=36. Suppose that Sindependent andandShow that Tindependent.andW₁ = V1 + V2 + V3-=37. Suppose that Sindependent andandShow that T =independent.{V1, V2, V3} is linearlyW3 = V3{W1, W2, W3} is linearlyW1 = V1 + V₂=38. Suppose that Sindependent andW2 = V2 + V3{V₁, V2, V3} is linearlyW3 = V2 + V3{W1, W2, W3} is linearlyW2 = V2 V3W1 = V2{V1, V2, V3} is linearlyW2 = V1 + V3W3 = V1 + V2 + V3Determine whether the set T = {W1, W2, W3} islinearly independent or linearly dependent.39. Suppose that the set S = {V₁, V₂} is linearlyindependent. Show that if v3 cannot be written asa linear combination of v₁ and v2, then{V1, V2, V3} is linearly independent.40. Let S = {V1, V2, V3}, where V3 = V₁ + V2.a. Write v₁ as a linear combination of the vectorsin S in three different ways.

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122 c. Show that the matrix Chapter 2 Linear Combinations and Linear Independence 25. A = 26. A = cannot be written as a linear combination of M₁, M2, and M3. In Exercises 25 and 26, for the given matrix A determine if the linear system Ax = b has a unique solution. 1 2 0 -1 0 3 21 2 M = 3 2 4 1 -1 4 0 2-4 In Exercises 27-30, determine whether the set of polynomials is linearly independent or linearly dependent. A set of polynomials S = {p1(x), p2(x), ..., Pn (x)} is linearly independent provided for all x implies that 0 3 3 1 C₁p1(x) + c2p2(x) + + Cn Pn(x) = 0 C₁ C₂ = = Cn = 0 27. p1(x) = 1 p2(x) = −2+4x² P3(x) = 2x P4(x) = -12x + 8x³ 28. p1(x) = 1 p2(x) = x P3(x) = 5+ 2x - x² 29. p1(x) = 2 p2(x) = x p3(x) = x² P4(x) = 3x - 1 30. p1(x)=x³2x² + 1 p2(x) = 5x P3(x)=x²-4 P4(x) = x³ + 2x In Exercises 31-34, show that the set of functions is linearly independent on the interval [0, 1]. A set of functions S = {f(x), f2(x),..., fn(x)} is linearly independent on the interval [a, b] provided c₁ f1(x) + c₂f2(x) + + Cn fn(x) = 0 for all xe [a, b] implies that C1 C₂ = = C₁ = 0 31. f₁(x) = сos лx ƒ₂(x) = sin лx 32. fi(x)= et f₂(x) = ex f3(x) = ²x 33. f₁(x) = x f₂(x) = x² ƒ3(x) = ex 34. f1(x) = x f2(x) = ex f3(x) = sin лx 35. Verify that two vectors u and v in R" are linearly dependent if and only if one is a scalar multiple of the other. 36. Suppose that S = {V1, V2, V3} is linearly independent and W₁ = V₁ + V2 + V3 and W3 = V3 Show that T = {W₁, W2, W3} is linearly independent. 37. Suppose that S = {V1, V2, V3} is linearly independent and W1 = V1 + V2 and W2 = V2 + V3 W3 = V2 + V3 Show that T = {W₁, W2, W3} is linearly independent. and W2 = V2 V3 38. Suppose that S = {V1, V2, V3} is linearly independent and W1 = V2 W2 = V1 + V3 W3 = V₁ + V2 + V3 Determine whether the set T = {W₁, W2, W3} is linearly independent or linearly dependent. 39. Suppose that the set S = {V₁, V2} is linearly independent. Show that if v3 cannot be written as a linear combination of v₁ and v2, then {V₁, V2, V3} is linearly independent. 40. Let S {V1, V2, V3), where V3 = V₁ + V₂. a. Write v₁ as a linear combination of the vectors in S in three different ways.