Problem 8. Consider the inner product space from Problem 4. Find an orthonormal basis for the subspace of C[-1, 1] spanned by functions h₁(x) = 1, h₂(x) = x and h3(x) = x².
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![Problem 8. Consider the inner product space from Problem 4. Find an orthonormal
basis for the subspace of C[-1,1] spanned by functions h₁(x) = 1, h₂(x) = x and h3(x) = x².](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F47d370c3-9e4b-442d-9a89-d591c5ced338%2Ffb531972-830e-480b-891f-eee977a05ea4%2Fuyxcskq_processed.png&w=3840&q=75)
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- O -2x (3) A. 7x XER 5x -{]-*-*-*-²} B. y - 7x - 5y - 4z Which of the following sets are subspaces of R³ ? C. ● Z F. X x + 3 x - 8 ● X •{:) -->0} D. y Z | XER} X = { ;] x + y + ² = 0 } E. X Z X {] Z - 3x + 8y = 0 & 6x2z =25. (a) Find all 2 ×x 2 matrices whose null space is the line 3x – 5y = 0. (b) Describe the null spaces of the following matrices: 6 2 C = 3 A B = D = 5 16. (Calculus required) Show that the following sets of functions are subspaces of F(-∞, ∞). (a) All continuous functions on (–0∞, 00). (b) All differentiable functions on (-∞, ). (c) All differentiable functions on (–o, 0) that satisfy f'+ 2f = 0. 9. Find the dimension of each of the following vector spaces. (a) The vector space of all diagonal n x n matrices. (b) The vector space of all symmetric n x n matrices. (c) The vector space of all upper triangular n x n matrices.*1. Which of the following are subspaces? Justify your answer in each case. (d) (x € R³ : x² + x² + x³ =1} (f) (x = R³: x² + x² + x² = -1} ---0-0-0- (h) (x ER³: x = 0 for some s, t = R}
- 1. Determine whether the set W =- X1,X, E R,x, = 0}} is a subspace of R' with the standard operations. Justify your answer.*2. For each pair of vectors in Exercise 1, calculate proj,x and proj,y. *1. Which of the following are subspaces? Justify your answer in each case. (d) (x € R³ : x + x + x² = 1) (f) (x = R³: x + x + x = -1} -0-0-0₁ +s + (h) (x = R¹ : x= for some s, 1 = R}19. Prove that the set of all vectors in R³that satisfy the equation x₁ = 3x2, is a subspace.
- Find the kernel and range of each of the following linear operators on P^3(the vector space of polynomials with degree less than 3):(a) L(p(x)) = xp′(x), where p′(x) is the derivative of p(x). Hint: Start with p(x) = ax2 + bx + c. (b) L(p(x)) = p(x) − p′(x).I.1. Find the adjoint of the matrix,then use the adjoint to find the inverse if possible 2.Find the Equation of (-4,0) and (4,4)3. Find the volume of the tetrahedron (2,0,0), (0,2,0), and (2,2,2).II.4. Determine whether the set of W is a subspace of R^3 with the standard operation. Justify your answer. a. b.5. Determine whether the set S is linearly independent or linearly dependent.a.b.c.Suppose y1 ( x), y2 ( x), y3 ( x) are three different functions of x. The vector space they span could have dimension 1, 2, or 3. Give an example of y1, y2, y3 to show each possibility.
- Consider the subspaces U = span{[4_0_1],[4 1 −4]} W = span{ [-5 3 3], [-5 33],[-5 4 −2]}Suppose that r1(t) and r2(t) are vector-valued functions in 2-space. Explain why solving the equation r1(t)=r2(t) may not produce all the points where the graphs of these functions intersect. Please Provide Unique Answer. Thank you!4) Let V be the vector space of polynomials of degree <2 and let p(x) = ax² + bx + c. Find the adjoint operator (T*@)(p) for T:V → V and o:V → R given by d) Tp(x) = x²p(x) + x³p'(x), @(p) = p"(1).