2. An inner product on a vector space V is any mapping ,-) V x V R with the following properties: i. Positive definiteness: (x, x) 0 for all x E V and (r, x) 0 if and only if x 0 ii. Symmetry: (x, y) = (y, ) for all r, yE V ii. Linearity: (axby, z) a(x, z)b(y, z) for all r,y, z E V and a, b e R. Show the following a) The dot product y= iyi is an inner product on the vector space R". i-1 To (b) The product (x, y) = x(t)y(t) dt is an inner product on the vector space of functions

Q: = Consider two bases B = {u₁, u₂} and B' = {V₁, V₂} for a certain vector space V. Assume that u₁ =…

A:

Q: Consider the inner product space (P2, (, )) with (p, q) = p(to)q(to) + p(t1)q(t1) + p(t2)q(t,),…

A:

Q: If x, y and z are vectors in an inner product space such that (x, y) = (x, z), then y = 2.

A: We are authorized to answer one question at a time, since you have not mentioned which question you…

Q: 2. Let R3 be a real vector space, and u = (u1, u2, u3), v = (V1, v2, v3) E R", then show that =…

A:

Q: Let x, y, z be (non-zero) vectors and suppose w = -12x - 6y + 2z. If z = 4x + 2y, then w = x+ A.…

A: given that  w= -12x- 6y+2z  and if z=4x+2y then  w= -12x-6y +2(4x+2y)  = -12x -6y +8x+4y w= -4 x -2…

Q: 6. Let V be a vector space over F and let u, v, we V, a, Be F. Which of the following is not a…

A: In this question, we will discuss about the properties of vector space.

Q: 6) Consider the following vector spaces with corresponding basis: V₁ =R³ Bv₁ = {e₁,e2, е3} Bw₁ = {1,…

A:

Q: Find a vector-valued function whose graph is the plane x+y+z== a. r(u,v)=ui-vj-(3-u-v)k b. r(u,v)=…

A: The equation of plane x+y+z=3

Q: If L is a line in 2-space or 3-space that passes through the points A and B, then the distance from…

A:

Q: Let U = ((1, 1, 0, 1), (0, 1, 2, 3), (2, 3, 2,−1)) and W = ((1, 3, 4, 4), (1, 1, 0, 2)) be sub-…

A: The given U=1, 1, 0, 1, 0, 1, 2, 3, 2, 3, 2, -1 and  W=1, 3, 4, 4, 1, 1, 0, 2 be a subspaces of ℝ4.…

Q: A line l passes through the point P(2,4, 3) and is parallel to the vector (3, 2, –3). | At what…

A: Proceed as shown below

Q: HIf vectors x, y, z, u in the real vector space p (the polynomials) are defined by x(t)=1, y (t)=t,…

A:

Q: 7. Let u = (2,1, −1) and v = (√√2, 1, 1). Find the decomposition of u into vectors that are parallel…

A:

Q: Find ProjS(u) where S is the space spanned by v1= (1/3)(2,1,2) and v2= (1/3)(-2,2,1). Find the…

A:

Q: 123. If F is a vector point function and a is a constant vector then show that i) V (a · F) = (a-V)…

A:

Q: Let / be the line through the point with coordinates (1,1,1) in the direction of the vector on [? O…

A:

Q: function V(x,y,z) = x² + xy + yz and a vector A = x² + xy + yzź are given. a) Find VV(x, y, z). b)…

A: Given function v(x,y,z) is

Q: H is the set of all vectors x = X2 in R³ with the property that X3 X1 + x2 = 0 and Xi – x2 + x3 = 0.…

A:

Q: Let R" be a real vector space, and u = (u1, u2, Un), v = (v1, V2, ..., Vn) E .... R", then show that…

A: For a real vector space, the inner product . , . satisfies the following properties a)  u+w , v =…

Q: 7. On a two-dimensional plane, sketch the vector |v>= such that it emanates from the coordinate…

A:

Q: 10) (6 points) Using the inner product (v, w) = 3v¡W1+5v2W2 with the vector space R?. a) Find a…

A:

Q: Use the function to find the image of v and the preimage of w. T(V1, v2) = (Vị + v2, V1 - V2), v =…

A:

Q: Find a vector-valued function whose graph is the plane x+y+z= 12, O r(u, v)=-ui-vj+(12-u-v)k O r(u,…

A:

Q: Let f,g be scalar functions, ủ be a vector function, and assuming sufficient differentiability. Then…

A:

Q: Let r E R and --(), v- (), -- (1). 3 u = V = W = 2 For which values r ER is the set {u, v, w}…

A:

Q: Let A(t) = f(t)i + f₂(1) and B(t) = g₁(t)i + g₂(t)},t = [0, 1], where f₁,2, 8₁, 82 are continuous…

A: First we use the condition of parallel vectors and then by intermediate value theorem we will find…