. Let a be a fixed vector in R³, and define W to be the subset of R³ given by T W = {x: a¹ x = 0}. Prove that W is a subspace of R³.

Q: Show that V is not a subspace of R°: V is the set of all (x, y, z) such that x + y +z = 3

A:

Q: Consider the R-vector space V-R2. Which of the following subsets of V is a subspace? The line 2x+y=1…

A:

Q: Let W be the set a of all vectors of the form | 0 [2 Is W a subspace of R³

A:

Q: 2 Let R¹ be the vector space over R with standard addition and scalar multiplication. Let X = {(x,…

A: X=x,y,z,w∈ℝ4: x-2y-3y-4z=0

Q: Let V = {(x, y, 2x – 3y)|x, yɛR}. Prove that V is closed for addition and scalar multiplication i.e.…

A:

Q: Let V be the vector space of real-valued functions on R. Let H be the set of all linear equations;…

A:

Q: Let V be a finite-dimensional vector space and let T: V -> V be linear. Show by Rank-Nullity Theorem…

A:

Q: Let V be an n-dimensional vector space. Define f: V→ R by f(v) = |v|". Let p,q E V.Find d fp(q).

A:

Q: Find a basis and the dimension of each of the following subspaces of R" и a. W1 = v: u+ v = 0 and u…

A: For (a), Now, we need to find the basis. Here, u+v=0⇒v=-u and u-2w=0⇒w=u2. So,…

Q: Let V be a finite-dimensional vector space, and let U, W be subspaces of V. Prove that (U + W)° C…

A:

Q: Prove that W1 = {(a,b) E R² : 4a – b= 0} is a subspace of R2.

A:

Q: Let V be the two-dimensional subspace of R4 spanned by (0, 1, 0, 1) and (0, 2, 0, 0). Write the…

A: To find v, we need to find the projection of u onto the subspace V, which can be found using the…

Q: Produce an example of an element of Co (R, R) which is not an element of the following vector…

A: Correct answer is f(x)= Sin(x) which is infinitely many times continuously differentiable but it is…

Q: Prove that for all divides natural numbers nt 7^-2^.

A:

Q: Let W be the set a of all vectors of the form | 0 2 Is W a subspace of R

A: The question is to show whether W is a subspace or not. We show that the sum of two vector does not…

Q: Letv be a vector Space and let V Vz rin Vnt V. ret w=span { vi, VE, n VnS. Prove W is a subspace of…

A: A necessary and sufficient condition for a nonempty subset W of a vector space V to be a subspace…

Q: Determine whether U = {(1, s, t)ls,t e R} is a subspace of R.

A: FOR ANY SUBSET U TO BE A SUBSPACE OF ANY VECTOR SPACE V IT IS NECESSARY THAT U HAS ZERO VECTOR IN…

Q: Consider the following vectors in R³: V1 = 1 Let V be the subspace spanned by {w₁, W2, W3}. Find a…

A: be the subspace spanned by

Q: Consider the vector space R 4 over the field of real numbers F = R 4. Let W = { ha, b, c, di | where…

A:

Q: et W be a subspace spanned by the u's, and write y as the sum of a vector in W and a vector…

A:

Q: Let W be a subspace of R" and suppose that there is a vector v in R" with v = Projwv, then which of…

A: Let us solve above.

Q: Let V be a finite dimensional vector space and W CV a subspace. Prove that W has a complimentary…

A:

Q: Let W be a subspace of R". A set of vectors {v1, ..., Úp} in W is defined to be a basis of W if {v1,…

A: Let ℝn be a vector space. Let W be a subspace of ℝn. Then,  is W={v1→,......,vp→}a basis. Definition…

Q: Let U and W be subspaces of a vector space V. Show that Un W is a subspace of V and that U + W =…

A: Since V is a vector space and U and W be subspaces of V. (i) We have to prove U∩W is a subspace of…