If a11 a12 b11 b12 А — and B= | b21 b22. a21 a22 are arbitrary vectors in R2x2, then the mapping (А, В) a11b11 + a12b12 + a2ı b21 + a22b2 defines an inner product in R2x2. Use this inner product to determine (A, B)), ||A||, ||B||, and the angle a A.B between A and B for -4 А — [ 2 and B= 3 -3 1 -1 (А, В) ||4| | ||B|| a A,B =

Q: Suppose that u, v, and w are vectors in an inner product space such that (u, v) = 1, (u, w) = 5, (v,…

A: A inner product space satisfies the following properties:1. Linearity:- (ax+by, u)=a(x,u)+b(y,u)2.…

Q: Find the dimension of the vector space.M2,3

A: We know that is a vector space over R. Consider the set where

Q: If i = (a, b, 0), ï = (a, 1, 6), and (1,1, 12) are vectors in R Determine the value of b so that and…

A:

Q: Give an example of a set of nonzero vectors in R3 that generates R3 but is not a basis for R3.…

A: Definition 1 : A basis β for a vector space V is a linearly independent subset of V that generates…

Q: 4. For vectors v and ū in an inner product space V, prove that u´ projzu and i are orthogonal.

A: v

Q: Let a1 and a2 be linearly independent vectors in R3, and let x be a vector in R2. (a) Describe…

A: Choose a1, a2 such that they are linearly independent in R2. Choose x ∈ R2 arbitrarily

Q: Suppose that u, v, and w are vectors in an inner product space such that (u, v) = 1, (u, w) = 5, (v,…

A:

Q: Show that, in an inner product space, there cannot be unit vectors u and v with <u, v > <…

A:

Q: Show that if S = {v1,v2,.., Vn} is an orthogonal set of vectors, then for any scalars a1, a2, ... ,…

A: If S is an orthogonal set of vectors then to deduce that for any scalars set T is also an orthogonal…

Q: Show that the vectors V1 = ( 1, 1, 2, 4 ), V2 = ( 2, -1, -5, 2 ), V3 = ( 1, -1, -4, 0 ) and V4 = (…

A:

Q: Find the dimension of the vector space.M3,2

A:

Q: 5. a) Prove that -2-M u-+ for all vectors in R". b) Explain parts a) geometrically.

A:

Q: Let y = and u = Write y as the sum of two orthogonal vectors, y, in the span of u, and y, which is…

A: Solution

Q: Prove that the set of vectors B = {(1,2,3), (3,2,1),(0,0,1)} is linearly independent and spans R³.

A:

Q: Let a = (2,3,6) and b = (3,4,2) be vectors.  Compute the cross product aXb

A:

Q: If [b1 b12 [b21 b22] a11 A = a12 and B= a21 а22 are arbitrary vectors in R2x2, then the mapping (A,…

A: We are given the inner products defined for the matrix, A and B and we need to find the values of…

Q: The two vectors u=(-2,3,1,4) and v= (1,2,0, -1) are orthogonal in the vector space R4 True False

A:

Q: at vectors （c1 - a1, c2 - a2）and （b1-a1,b2-a2）are linearly depend

A:

Q: Find an orthogonal basis for A1, A2, A3 (i.e., find 3 mutually perpendicular vectors u1, u2, u3 with…

A: The solution are next step

Q: Suppose that u, v, and w are vectors in an inner product space such that (u, v) = 1, (u, w) = 6, (v,…

A:

Q: Suppose that u and v are vectors in 3-space such that u ||= 5 and u · v= 7. Find: (a) u · u = (b) v…

A:

Q: Evaluate the expressions in. ||2u - 3v + w||

A: Given,

Q: 3. Let v, and v2 be vectors in an inner product space V. Show that ||v + v2||? < (||Vvl+ ||v2||)?

A:

Q: If [a11 a12 A = a21 and B = a22 [b11 b12 b21 b22] are arbitrary vectors in Rx2, then the mapping (A,…

A:

Q: Determine whether the given vectors (1, 1,-1, 1), (1,-1, 2,-1), (3, 1, 0, 1) in R* are lincarly…

A: Given vectors are: (1, 1, -1, 1) , (1, -1, 2, -1) , (3, 1, 0, 1) We have to determine whether the…

Q: Let a = (7, 10, 1) and b = (10, 3, 6) be vectors. Compute the cross product %3D ахЬ.

A:

Q: Write x as the sum of two vectors, one in Span fu1,u2,u3 and one in Span fu4. Assume that fu1,...4is…

A: Let x = v + wWhere v = a1u1 + a2u2 + a3u4        and w = a4u4.

Q: Calculate a projection of vector B on A and prophecy of vector A on B. A = 4i - 3j + 2k.     B = 3i…

A:

Q: Show that the vectors u1=(1, 1, 1), u2=(1, 2, 3), u3 =(1, 5, 8) span R3

A:

Q: Consider the set V of all real triples of the form (a, b, c) where c = a - 2b under the usual vector…

A:

Q: Suppose u, v, and w are vectors in an inner product space such that (u, v) = 1, (v, w) = 0, (u, w) =…

A: Use the values given above for substitution after simplifying the problem.

Q: Find the scalar and vector projections of b onto a. a = (4,7,-4) b =(4,-1, 1)

A:

Q: -2 Show whether or not the vector bs is a linear combination of the vectors a, = az = and az = that…

A:

Q: If ã = <!,2,37 and b=<5,0 -17, find the vector pojection of %3D %3D ŏ onto à.

A:

Q: Find the scalar and vector projections of b onto a. = (3, –6, –2), b = (3, 3, 2) a

A:

Q: Show that if u and v are vectors in an inner productspace that satisfy the Pythagorean law||u + v||2…

A:

Q: : If v, v1, Vz are 3 vectors in a vector space V such that v – 9vi – 3v2 = 0. Then, v belongs to…

A:

Q: Which of the given sets of vectors in R3, with the standard inner product, are orthogonal,…

A: a) Given, 01-1,011,200 Consider the vectors, u=0,1,-1v=0,1,1w=2,0,0 Now find the inner product of…

Q: determine if the vector v is a linear combi­nation of th e remaining vectors.

A:

Q: Let 12, v, w be vectors In an inner product space such that //u|| =5, ||v||=5, and (u, v) =-2. Use…

A:

Q: When the set of vectors {u1, u2, . . . , un} is linearly independent and the set {u1, u2, . . . ,…

A: We have given; The set of vectors {u1, u2, . . . , un} is linearly independent and the set {u1, u2,…

Q: Suppose that u and v are vectors in 3-space such that |u||= 5 and u · v = 7.

A:

Q: If [b11 and B = b12 a11 a12 A a21 a22 b21 b22 are arbitrary vectors in RX2, then the mapping (A, B)…

A:

Q: ): If U, U₁, U₂ are 3 vectors in a vector space V such that 13v₁ = v + 7v₂. Then, u belongs to…

A: It is provided that v, v1, v2 are three vectors in the vector space: V and 13v1=v+7v2. We need to…

Q: 9. Determine if the vectors å = [-1, 2, 1], b = [-2,4, 2], and ở = [12,8, 1] are coplanar. %3D

A: given vectors a=[-1,2,1] b=[-2,4,2] c=[12,8,1]

Q: Suppose that u, v, and w are vectors in an inner product space such that (u, v) = 1, (u, w) = 5, (v,…

A:

Q: suppose that u, v, and ware vectors in an inner product space such that  &lt;u,v&gt; = 1,…

A: We have given u,v=1,u,w=5,v,w=0,u=1,v=3,w=2  We have to calculate 2u-3v+w

Q: Suppose that u, u and w are vectors in a real inner product space such that (u, w) = 16, (v, w) = 5,…

A: Instead of u¯,v¯,w¯ , we are writing u,v,w. (for sake of convenience). Given: (i) &lt;u,w&gt;=16(ii)…

Q: Find a vector b such that compab = 2. If a = ⟨3,0,−1⟩

A:

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.