6. Let B= {(v,,V½, v½ } be a basis for R'. Use the Gram-Schmidt process to find an orthogonal basis. Do not normalize the vectors.

Use the inner product u,v=2u1v1+u2v2 in R2 and Gram-Schmidt orthonormalization process to transform {(2,1),(2,10)} into an orthonormal basis.

Take this test to review the material in Chapters 4 and 5. After you are finished, check your work against the answers in the back of the book. Prove that the set of all singular 33 matrices is not a vector space.

Let B={(0,2,2),(1,0,2)} be a basis for a subspace of R3, and consider x=(1,4,2), a vector in the subspace. a Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B. b Apply the Gram-Schmidt orthonormalization process to transform B into an orthonormal set B. c Write x as a linear combination of the vectors in B.That is, find the coordinates of x relative to B.

Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors {vu,wv,uw} is linearly independent or linearly dependent.

Let v1, v2, and v3 be three linearly independent vectors in a vector space V. Is the set {v12v2,2v23v3,3v3v1} linearly dependent or linearly independent? Explain.

14a. Show that the vectors w₁ = (0,2,0), W₂ = (3,0,3), W3=(-4,0,4) form an orthogonal basis for R³ with the Euclidean inner product and use that basis to find an orthogonal basis by normalizing each vector. b. Express vector u = (1,2,4) as a linear combination of the orthonormal basis vectors obtained in part (a)

Let V = Polyz be the vector space of polynomials of degree 2 or less and let W = Poly, be the vector space of polynomials of degree 2 or less. Let B be the following basis for V and let B' be the following basis for W. B = ((1+1x + Oz²), (0+ 0x + (-1)x²), (0+1x + 0x²)) B' = ((-1+ 2x +(-1)x²), (0+ læ + 0x?), (1+ (-2)x+0x²)) Suppose that T :V → W is a linear map and that the matrix associated to T with respect to the bases Band B' is: -2 -1 -2 [T]g'+8 = -4 -1 -3 -1 -1 Find the value of T ((0+ læ + (-2)æ?)). Write down the values below if the answer is (a1 + (a2)x + (a3)a²) aj = a2 = az =

Consider linear mapping A E L(Vn) and two basis X, Y of Vn. Select all true statements. Let Y be a basis obtained from Y by swapping i-th and j-th vector. Then matrix * A' differs from * A° by swapping i-th and j-th row. Let X be a basis obtained from X by swapping i-th and j-th vector. Then matrix * A' differs from A' by swapping i-th and j-th row. Let X be a basis obtained from X by swapping i-th and j-th vector. Then matrix * A' differs from A³ by swapping i-th and j-th column. Let Y be a basis obtained from Y by swapping i-th and j-th vector. * A by swapping i-th and j-th column. Then matrix * A' differs from