Let V be a K-vector space and U1, U2, U3 three sub-vector spaces of V. Show that: dim(U1) + dim(U2) + dim(U3) = dim(U1 + U2 + U3) + dim((U1 + U2) ∩ U3) + dim(U1 ∩ U2) whereby dim stands for dimension

Q: Let V be a vector space, let c be a scalar and let u, v, w be vectors in V and ca scalar. Which of…

A:

Q: If V is a vector space then show that cV=V.

A: Vector space: A non-empty set V closed with two binary operations called addition and scalar…

Q: 6. (a) Let u, v, w be three linearly independent vectors in Euclidean Space R^7. Determine a value…

A: we have to find the value of for the linearly dependent and linearly independent

Q: State whether the following statement is true or false: A vector space may have two or more zero…

A:

Q: Let (V, (, )) be an inner product space and let f, g be vectors in V. if || f|| = 4, ||| 5 and || f…

A:

Q: Let 7 = -27+ 7+2k and w = 3 7 +23- k. (a) Compute the vector projection of w onto , proj- w. (b)…

A:

Q: Consider set V = { (x, y ), x,yER} with the following operations: addition (x1, y1) + (x2 , Y2) = (2…

A:

Q: 1. Is V = why not? 1:0 20 and b≥( a vector space? Draw the graph and explain why or

A: By the answering guidelines we have to do first question is multiple questions are given so I am…

Q: If V is vector space, with u = (u, , U2) and v = (v, , v2) with u +v = (u,+v, , u2+v2) and ku =…

A: u=(u1,u2) , ku=(3u1,0) k=2, u=(-1,2)

Q: 2) Write the vector v as a sum of proj span{.…...} (v) and proj spo span{m} orthogonal set V₁,..., V…

A:

Q: Determine the element of V that represents the zero vector 0 (i.e., determine the “value” of 0 in V…

A:

Q: (2) Let M₂×2 be the vector space of all 2X2 matrices, and defined T:M 2x2 (A)=A+AT. where A= a [cd]…

A: Given the  vector space of all  matrices, and defined , such that  where .The aim is to describe the…

Q: 2. fu = and v = v1, v2, V3 > are two vectors in 3-space show k that u x v= u1 U2 U3 V1 V2 V3

A:

Q: Show that the set of all second order polynomials, f(x) = ax² + bx+c, is a vector space.

A:

Q: Show that W is a vector space.

A:

Q: Determine whether the following sets with the corresponding vector addi- tion and scalar…

A:

Q: Show that the function (V1, V₂) = 2w1W2 + x1x2 - Y1Y2 +2122 defined for any vectors V₁ = (x1, y1,…

A:

Q: Consider the vector space 2 with the inner product rgk)=x+1,then %3D

A:

Q: If scalar multiplication and addition is redefined on R² in the following way, show that it is NOT a…

A:

Q: Let V = R2, defined the addition by: (u1 tu2) + (v1 + vz)=( U1 +v1 , Uz tv2 + 1) With standard…

A: Given that the vector addition on V=ℝ2 is defined by           u1,u2+v1,v2=u1+v1,u2+v2+1 with…

Q: Supp0se A = {u1, u2, u3} be a set of linearly dependent vectors. Let A be in an inner product space…

A: Please check the next step for details

Q: : Let V be the set of all ordered pairs of real numbers (u1, u). Consider the following addition and…

A:

Q: Suppose {v₁, ..., vr} is linearly independent but does not generate the vector space V. Show there…

A:

Q: ive an example of a space is not vector space

A:

Q: Let V1, V2, V3 be vectors in R³ given by 1 »-(;) · ~ -(:). = 6 V2 = 2 2 0 The set {V1, V2, V3} is…

A:

Q: Let P, is the vector space of polynomials of the degrees2.Use coordinate vectors to that the…

A: The main objective is to check {p1,p2,p3} linear independent and basis of P2.

Q: w that the space V = {(x1, x2, x3) ∈F3

A: To prove that the space V=(x1, x2, x3)∈F3|x1+2x2+2x3=0 forms a vector space.

Q: If {v1, V2, V3} is an orthogonal set of nonzero vectors in an inner product space V, then prov that…

A: This question is related to linear algebra (inner product space).

Q: Does the empty set form a vector space?

A:

Q: Show that there do not exist scalars c₁, c₂, and C3 such that c₁(1,0, 1,0) + c2 (1, 0, 2, 1) +…

A:

Q: Does the set of all functions that vanish at x = 0 and x = L form a vector space? If it does,…

A: 1) Commutativity of addition: Let f(x) and g(x) be functions in V.Then, .Therefore, addition is…

Q: Give two examples of finite dimension vector spaces.

A:

Q: Question 35 Let B = {b1, b2} and C = {C1, c2} be two bases for a vector space V. You are given that…

A:

Q: Give an example of 3 vectors spaces that are not Rn. Explicitly state the definition of addition and…

A:

Q: Any given vector space may have distinct inner products.  True  False.

A: Given :- Any given vector space may have distinct inner products. To find :- The given statement is…

Q: Show that the set of polynomials of degree 3 form a vector space. P(x) = ax³ + bx² + cx+d, a, b, c,…

A: We can show that the set of all polynomials of degree 3, where P(x) = x^3 + bx^2 + cx + d (a, b, c,…

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.