Let r= (11, 12) and y=(41,42) be vectors in the vector space Cover C, and define...
Question 3. Let 3 5/' and for x(2),y -(,) ER2 define (a) Show that the assignment (x, y) > (x,y) defined ın (1) us an nner product [10 marks (b) If a - (1,-1) and b - (1,1), then show that the vectors a and b are lınearly ndependent but they are not orthogonal with respect to the inner product n (1) 3 marks] (c) Given the vectors a and b in (b), the set (a, by is hence a...
QUESTION 2 Consider the vector space R3 (2.1) Show that (12) ((a, b, c), (x, v, z))-at +by +(b+ c)(y + z) is an inner product on R3 (2.2) Apply the Gram-Schmıdt process to the following subset of R3 (12) to find an orthogonal basis wth respect to the inner product defilned in question 2.1 for the span of this subset (2.3) Fınd all vectors (a, b, c) E R3 whuch are orthogonal to (1,0, 1) wnth respect to the...
Let uj = [1,1,1]* and u2 = [1,2,2]t be vectors in R3 and V be the vector space spanned by {u1, U2}. a. 6pt Use Gram-Schmidt orthogonalization to find an orthonormal basis for V. b. 4pt Let w = [1,0,1)+. Find the vector in V that is closest to w.
Find a basis for the vector space W spanned by the vectors$$ \overrightarrow{v_{1}}=(1,2,3,1,2), \overrightarrow{v_{2}}=(-1,1,4,5,-3), \overrightarrow{v_{3}}=(2,4,6,2,4), \overrightarrow{v_{4}}=(0,0,0,1,2) $$(Hint: You can regard W as a row space of an appropriate matrix.)Using the Gram-Schmidt process find the orthonormal basis of the vector space W from the previous questionLet \(\vec{u}=(2,3,4,5,7)\). Find pro \(j_{W} \vec{u}\) where \(\mathrm{W}\) is the vector subspace form the previous two questions.
5. Let X, Y be ordered bases for the vector space R", and define M to be the transition matrix from Y to X. Let A be the matrix formed by using the vectors of X as the columns. Prove or disprove that AM must be the matrix formed by using the vectors of Y as the columns.
2 over R. Define U g-Ло íj fg for f,0€ y 5. Let V be the vector space of polynomials of degree a. If U is the subspace of scalar polynomials, find U b. Apply Gram-Schmidt to the basis1, t,t2) of V. c. If a E R find ga E V with (f, ga)(a) for all f E V
#8. Let W be the subspace of R3 spanned by the two linearly independent vectors v1 = (-1,2,2) and v2 = (3, -3,0). (a) Use the Gram-Schmidt orthogonalization process to find an orthonormal basis for W. (b) Use part (a) to find the matrix M of the orthogonal projection P: R W . (c) Given that im(P) = W, what is rank(M)?
Q3 14 Points Consider the vector space P2(R). Let ri, r2, 13 be 3 distinct real numbers and d1, A2, A3 be three strictly positive real numbers. Define (p(x), g(x)) = Li-1 aip(ri)q(ri) Q3.3 5 Points Let rı = -2, r2 = 1, r3 = 2, a1 = 1, a2 = 2, a3 = 3. Apply the Gram-Schmitt Orthogonalization process to the basis (1, x, x2). Write monomials in descending order of their exponent, x^n for æ" and a/b form....
In the vector space R, let 8 {(1,3,0), (1, -3, 0), (0, 2, 2)}. (a) (6 points) Show that y is a basis of R3. (b) (7 points) Find the matrix [I,where I is the identity transform R3 R3 (c) (7 points) Using the matrix [I, convert the vector (r, y, z) into coordinates with respect to y instead of B. In other words, find ((x, y, z)] {(1,0,0), (0, 1,0), (0,0, 1)} be the standard basis, and let
3. Consider the following vectors, where k is some real number. H-11 Lol 1-1 a. For what values of k are the vectors linearly independent? b. For what values of k are the vectors linearly dependent? c. What is the angle (in degrees) between u and v? 4. Here are two vectors in R". Let V = the span of {"v1r2} a. Find an orthogonal basis for V (the orthogonal complement of V). b. Find a vector that is neither...