3. Let R3 be equipped with the inner product (x, y) = AX Ay, where A...
3. Let R3 be equipped with the inner product (x,y) = Ax. Ay, where A is the matrix shown below: TO A=13 LO -4 2 0 2 1 5) a.) (5 points) Let v = (1,-1,3). Find ||v||. b.) (5 points) Let x = (2,3,0) and y = (-3,2,1). Are x and y orthogonal in this inner product space? Justify your answer.
3. Let R be equipped with the inner product (x,y) = AX Ay, where A is the matrix shown below: TO-4 21 A = 3 2 LO 0 5) a.) (5 points) Let v = (1,-1,3). Find || V || 1 b.) (5 points) Let x = (2,3,0) and y = (-3,2,1). Are x and y orthogonal in this inner product space? Justify your answer
3. Let V be the space of n X 1 matrices over C, with the inner product (X\Y) = YGX (where G is an n x n matrix such that this is an inner product). Let A be an n x n matrix and T the linear operator T(X) = AX. Find T*. If Y is a fixed element of V, find the element Z of V which determines the linear functional X + Y*X. In other words, find Z...
8. More generally, let X be any infinite-dimensional vector space equipped with an inner product ,) in such a way that the induced metric is complete. In particular, there is a norm on X defined by and the metric is given by d(r, y) yl Let A denote the unit ball A x E X < 1} We know that A is closed and bounded essentially from the definitions. Show that A is not compact. (Hint: Construct a sequence xn...
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...
4. Consider the vector space V = R3 and the matrix 2 -1 -1 2 -1 -1 0 2 We can define an inner product on V by (v, w) = v'Mw. where vt indicates the transpose. Please note this is NOT the standard dot product. It is a inner product different (a) (5 points) Apply the Gram-Schmidt process to the basis E = {e1,e2, e3} (the standard basis) to find an orthogonal basis B. 4. Consider the vector space...
Let V be a finite dimensional vector space over R with an inner product 〈x, y〉 ∈ R for x, y ∈ V . (a) (3points) Let λ∈R with λ>0. Show that 〈x,y〉′ = λ〈x,y〉, for x,y ∈ V, (b) (2 points) Let T : V → V be a linear operator, such that 〈T(x),T(y)〉 = 〈x,y〉, for all x,y ∈ V. Show that T is one-to-one. (c) (2 points) Recall that the norm of a vector x ∈ V...
6. Let L be the linear operator mapping R3 into R3 defined by L(x) Ax, where A=12 0-2 and let 0 0 Find the transition matrix V corresponding to a change of basis from i,V2. vs) to e,e,es(standard basis for R3), and use it to determine the matrix B representing L with respect to (vi, V2. V
1). Let V be an n-dimensional inner product space, let L be a linear transformation L : V + V. a) Define for inner product space V the phrase "L:V - V" is an orthogonal transforma- tion". b) Define "orthogonal matrix" b) If v1, ..., Vn is an orthonormal basis for V define the matrix of L relative to this basis and prove that it is an orthogonal matrix A.
c and d only 2. Consider the vector space R3 with the standard inner product and the standard norm |x| x, x) Use the formula for projection given in Chapter 5, Section 4.2 of LADW to find the matrix of orthogonal projection P onto the column space of the matrix -) 1 1 A = 2 4 (a) What is the projection matrix P? (b) What is the size of P? (c) Since the dimension of the column space of...