If (x1,y1) + (x2,y2) = (x1x2,y1y2) and  c(x1,y1)= (cx1,cy1) are vector spaces, verify if both are a vector space property and identify the properties that it will fail. (Use the 8 theorems)

Q: Find one voctor in R' that spans the intersoction of U and W where U is the xy-plane-that is, U =…

A:

Q: 3. Show that in a normed space X, vector addition and multiplication by scalars are continuous…

A:

Q: Use proof by contradiction to prove the statement.In a vector space, the zero vector is unique.

A:

Q: Find a linear mapping G that maps [0, 1]×[0, 1] to the parallelogram in the xy-plane spanned by the…

A: A linear mapping G that maps [0, 1]×[0, 1] to the parallelogram in the xy-plane spanned by the…

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: Determine whether the vectors u = 2 – x + 3x?, v = 4 +x + 2x?, and w = 8 - x + 8x² span P2 (the set…

A: The set {p1,p2,p3} will span p2 if and only if they are independent  so, here we test if the vectors…

Q: Find the norm of the vector U(2,-6,1)

A: See the solution below.

Q: Suppose V1, ..., Vn are vector spaces then does L(V1 x...x Vn, U) have the same dim with L(V1, U)…

A:

Q: Give an example with values of a nonstandard operation for M2*2 a vector space that fulfills all…

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

A:

Q: ) Define what it means for a set of vectors {v1, . . . , vn} from a real vector space V to be…

A: we have to define what it means for a set of vectors {v1, . . . , vn} from a real vector space V to…

Q: Determine together with standard operations, is a vector space. If not identify one of the property…

A: set of all matrices with a fixed form is always a vector space  a) M2,4  is a vector space of 2x4…

Q: The dimension of the vector space ii2×3 is equal to 6. Select one: True O False

A:

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

A:

Q: Explain the term A Basis for a Vector Space.

A:

Q: Find a bases for the vector space V spanned by vector w = (1,1,0),w, =(0,1,1), w; =(2,3,1) and w,…

A:

Q: Show that W is a vector space.

A:

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: Are the vector spaces R4, M2,2, and M1,4 exactly the same? Describe their similarities and…

A: We have to determine if the vector spaces, are exactly the same or there is any similarity or…

Q: Can a vector in a vector space R -4 - 2 be 2. span by -1 2 イ

A: Span of vectors.

Q: Determine whether the vectors u = 8x² span P2 (the set of polynomials of degree 2 or less). If not…

A: Here given that P2 has dimension 2 , the set { P1,P2,P3} will span P2 , if and only if they are…

Q: Explain the term Dimension of a Vector Space.

A:

Q: Let (V, (, )) be an inner product space and let v, w be vectors in V. If (v, w) = 7 and w = 5, then…

A:

Q: : The dimension of the vector space Pa is equal to: Select one: 4 2 none of the others

A: Any vector in the vector space P4 is a linear combination of this set of polynomials: {1, t, t^2,…

Q: Describe the zero vector and the additive inverse of a vector in the vector space.P8

A: The given vector space is P8 .

Q: In words, explain why V is closed under addition and scalar multiplication. Since addition on V is…

A: According to the given information, first it is required to explain why V is closed under addition…

Q: Let v1 , v2 , . . . , vn be linearly independent vectors in a vector space V. Show that v2, . . . ,…

A: If the given vectors are linearly independent in the vector space V. Then it follows that

Q: What are the coordinates of the vector 0 in the Euclidean plane that satisfies condition - "There…

A: The co-ordinates of the 0 vector in the Euclidean plane is

Q: Determine whether the set, together with the standard operations, is a vector space. If it is not,…

A:

Q: how can one check that a set with two operations, addition and scalar multiplication, is a vector…

A: One can check for a set to be a vector space by checking all the above axioms. If a set V with two…

Q: What is the dimension of the vector space spanned by the vectors v1, v2? v1 = (6, -2, 4) and v2 =…

A: v1 = (6, -2, 4)  v2 = (14, 2,8)  There does not exist any real number c such that v1 = c v2 hence v1…

Q: Describe the zero vector and the additive inverse of a vector in the vector space.M2,3

A:

Q: Describe the zero vector (the additive identity) of the vector space. P3 x + ]+² +

A: We have to describe the zero vector (the additive identity ) of the vector space P3.

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: (b). Let V = R? = {(r,,1)| plication. Is it a vector space? IER} with usual addition and scalar…

A: Is it vector space

Q: Give two examples of finite dimension vector spaces.

A:

Q: x1 ,x2 ,...,xn  span a vector space V, then they are linearly independent. Select one: True False

A: We have to check

Q: Show that the set of all pairs of real numbers (x, y) with the operations (X1, Y1) + (x2, Y2) = (x1…

A:

Q: Let V = Rº. Give a definition of addition + or scalar multiplication on V that makes (V,+,-) unable…

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: Suppose there is a collection of three or more two- dimensional vectors. Provide an argument…

A: Collection of two dimensional vectors mean those vector having i and j components.

Q: not vector spaces

A:

Q: determine projection of the vector b = -1 1 1 on the space generated by the vectors {--E)--[ -[+]}…

A:

Q: If V1, V2, V3, V4, and W are all vector spaces.  Are L(V1x...xV4, W) and L(V1, W) x...x L(V4, W)…

A: Given that: V1,V2,V3,V4,W=Vector space The objective is to verify the given statement. Using the…

Q: True or False: a) Vector space Z2³ contains 8 vectors.

A:

Q: Verify that P,(R) = Eax | a, eR}, with addition and scalar multiplication !! 1-0 defined by Ź(a,…

A: let  k1  and k2  be two scalar  and  ∑aii=02xi   ∈P2ℝif P2ℝ is vector  space with given operation…

Q: Prove that the relation of orthogonality in.an inner product space is symmetric.

A:

Q: Show that (x, y) = 4x1y1-7x2Y2 for vectors x = (x1, x2) and y = (V1 Y2) in R2 defines an inner…

A:

Q: now that Reis Banach f|xl|= | ( 2 | ², 1² ) / 2 j=1 and vector space. 2h is complete.

A: We have given Rn andx=∑j=1n  ξiq12. We have to show that Rn is Banch space, vector space and…