d) Prove that ||vxw||²+|v-w[? = ||v||²|w||² for all vectors v, w in R3

Q: Two vectors u and v in space are if and only if u X v= 0

A: Given vectors in space are u and v The vectors cross product is u x v = |u| * |v|* sin(angle between…

Q: Prove that for any non-zero vectors a, b, a · b = |a||b| cos(0), where 0 is an angel between a and…

A: Consider the n -dimensional vectors a and b , where n≥2 . The vectors a , b and a−b can be…

Q: Find the value of k such that vectors x

A:

Q: What should k be if the vectors x=(1,1,k) y=(1,3,1) z=(k,1,3) are in the same plane in the Euclidean…

A: Three vector a1,a2,a3,b1,b2,b3 and c1,c2,c3 will be in euclidean space ℝ3 If  a1a2a3b1b2b3c1c2c3=0…

Q: Question2 Let L: R R be defined by: L Find all possible 一-0 vectors u such that L U2 33

A:

Q: 5. Determine the number of vectors (x1,...,x„), such that each x; is either 0 or 1 and i=1

A: (x1,x2,...,xn) is a vector with xi is either 1 or 0.  Now, ∑i=1nxi ≥ k implies at least k of the…

Q: The set of all x ɛR ( means: x belongs to R ) for which the vectors : [1, x, 0] , [ 0, x² , 1] ,[0,…

A: A set of n vectors of length n is linearly independent if the matrix with these vectors as columns…

Q: Let v, w, and a be nonzero vectors such that v · a = w · a.Is it true that v = w? Either prove this…

A: Given: v, w and a is non zero vectors, such that: v·a=w·a To prove: v=w

Q: If (a, b) is a local maximum of f, then D, f(a, b) = 0 for any unit vector u e R?. True False

A: Question: if (a,b) is a local maximum of f, then Du→f(a,b)=0 for any unit vector u→∈R2.

Q: Prove that u is orthogonal to v - proju(v) for all vectors u and v in Rn, where u ≠ 0.

A:

Q: 4) Suppose x and y are vectors in R". Prove a) r+ >f -x->f =4(x• y) b) If x and y are orthogonal…

A:

Q: (b) For any arbitrary vectors u, v and w, prove that (u x v) x w = u x (v x w) if and only if (u x…

A:

Q: 3/ Given a vector x, show that the infinity norm defined by || X 1100 = max | X₁ Do Ksign

A: Given a vector X A norm is a function such that the following conditions hold ∥x∥≥0 ∀x∈Rn ∥x∥=0 if…

Q: Let r = (2,-i, 1), and y = (-i, 3,1 + i) be vectors in C3. Compute (x, y), ||x||, |ly|l, and…

A:

Q: Decompose V into in two vectors V1 and V2, where V1 is parallel to W and V2 is orthogonal W. V=2i+j…

A:

Q: _3. Let x be a convex combination of {y1, . . , yk }. Assume in turn that each y; is a convex…

A:

Q: If x = (x1,x2)T, y = (y1,y2)T, and z = (z1,z2)T are arbitrary vectors in R2, prove that (a) xTx ≥ 0…

A: The objective is to prove the above expressions.

Q: ): Let v = 3 be two vectors in R. and v2 %3D Show that v = belongs to span{v1, v2}. 62

A: We know that if vector "v" is linear combination of "v1" and "v2" then vector "v"belongs to the span…

Q: 4. Prove that, for any vectors A, B and C, that a necessary and sufficient condition А x (Вx С) %3…

A: The solution is:

Q: Let u = (−1, 1), v = (0, 1),and w = (3, 4). Find the vector x that satisfies u−2x = x−w + 3v.

A: u=-1, 1,  v=0, 1  &  w=3, 4

Q: Suppose we have a set of vectors v_1...v_n such that v_1 could be written as a linear combination of…

A:

Q: Prove that ||v×w||2 +|v.w|2=||v||2||w||2 for all vectors v,w in R3

A:

Q: Two vectors u and v in space are if and only if u .v=0

A:

Q: . Prove that for any two vectors d and b in R³, there exist two vectors bí and b such that: T = 6 +…

A: The vectors a→ and b→ lie in the 3 dimesional axes.Any vector is a sume of two vectors.Thus,…

Q: Show that 0 x a =0 = a x 0 for any vector a in V3

A: lets take vectorv=(x,y,z) and   0=(0,0,0)

Q: Please solve this linear algebra question for me.

A:

Q: f || x, is Infinite norm of vector x on R then Al, = max Ax|, Select one: True False

A: We have to check whether the statement, "Given x∞ is infinite norm of vector x on Rn then…

Q: Ex: Show that if v, u, w are any vectors, then (H.W.) ux (v x w) + v x (w x u) +w x (u x v) = 0.

A: Dot Product Property :- a.b = b.a = a*b*cosθ

Q: 3. If x and y are two n-vectors, then prove that 1 (a) |x*y|< ]], \»l, - +=1 (Holder inequality) 1…

A: Given that x and y are two n-vectors. We have to prove the following :                          (a)…

Q: If V is the space of all vectors in R'that are orthogonal to W= Span{(1,0,2,1). (0,0,1,5), then…

A: Let us assume that u=x1, x2, x3, x4 represents all those vectors that are orthogonal to…

Q: If x, y are two no vectors, then prove that |xty | || X lle y llq by = + // use Holder inequality.

A:

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: Consider the set V of all vectors of the form (x y 2) where x, y and z are real numbers 3. Show that…

A:

Q: Prove that two non-zero vectors u and v in n-space are perpendicular if and only if u • v = 0

A:

Q: Use the Wronskian to determine if the function vectors 7,= x? + e* and 72= e²x are linearly…

A:

Q: Prove that for any vectors u, v, and w in R2 or R3, we have u ·(v + w) = u · v + u · w.

A: Let u=u1, u2, u3  ,  v=v1, v2, v3  &amp;  w=w1, w2, w3 be the vectors. Now, v+w=v1, v2, v3+w1, w2,…

Q: Let x = (2,1+i, i) and y = (2−i, 2,1+2i) be vectors in C3. Compute &lt;x, y&gt;,||x||, ||y||, and…

A:

Q: 1. Let u and v be vectors in R". Suppose ||u||2 = 2, ||v||² = 1, and u·v = 1. Compute

A:

Q: 7) Find all vectors v in F such that 2v + ) -)

A:

Q: Let W be the set of all vectors of the form t - s Find vectors u and v in R' such that W = span {u,…

A:

Q: For any vectors U, V and scalars a, b, prove that (aU) x (6V) = ab(U × V) (a) (b) (aU) - (bV) = ab(U…

A:

Q: 1. Do the vectors 日 span R (over R)? Justify your answer (using methods and results that we have…

A:

Q: Two vectors u and v in space are if and only if u .v = 0

A: Condition of dot product.

Q: 2. Determine all real numbers x e R for which the vectors (x, 0, 1), (2, x, 3), and (4, 5, 6) ar…

A:

Q: In Rn with inner product{x, y} = xTyderive a formula for the distance between two vectorsx = (x1, .…

A: Take two vectors x and y in an inner product space x=x1,…,xn and y=y1,…,yn then x,y=xTy

Q: Suppose that the set A={X₁, X2..., Xn}, X₁ ER" spans the entire space R", i.e. for any arbitrary…

A: Given: The set A=x1,x2,...,xn, xi∈ℝn spans the entire space ℝn, i.e. for any arbitrary n-vector…

Q: If x = (x1, x2)T , y = (y1, y2)T, and z = (z1, z2)T arearbitrary vectors in R2, prove that xTy = yTx

A: x = (x1, x2)T , y = (y1, y2)T are arbitrary vectors in R2 .

Q: b/Let u and v be a nonzero vectors in 3- space, and let e be the angle between these vectors when…

A:

Q: Consider the collection of vectors {v1, v2, V3}, where v1 = -3v2 + 7v3. Is the collection linearly…

A:

Q: If i • ( or v• (u x w) or w • (u x 7) equal to zero, then the 3, and w vectors u. are coplanar.…

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.