Let and V=M₂x2(F). W₁ = {(2) €V: a,b,cEF}. 0 { (-ai) € V: ab€ a, W₂ 1 F}. Prove that W₁ and W₂ are subspaces of V, and find the dimensions of W1, W2, W₁ + W2, and W₁ W2.

Q: Determine if the set W of vectors (x1 , x2) in R such that xr? + x2 = 1 is a subspace of R’.

A: W is not a subspace of R²

Q: 6. Let Vi and V2 be subspaces of R" such that Vi C V2. Show that any vector v E R" can be uniquely…

A:

Q: Consider V = R^3 and U consists of those vectors each of whose first component is non-negative.…

A:

Q: Let u = (3, 2, x) and v = (2x, 4, x). (a) Find all possible values of x that make u and v…

A: Given thata)We have to find all possible values of x that make u and v orthogonal.b)We have to find…

Q: Prove that the set of vectors of the form (a, b, c) where b = a +c is a subspace of R³.

A: According to the given information, it is required to prove that:

Q: 1. Determine whether w - {[:]•r:a-4-0} X1 E R? : : x1 – 23 = X2 is a subspace of R?.

A:

Q: Let W be the subspace spanned by the u's. Write y as the sum of a vector in W and a vector…

A:

Q: 19. Prove that the set of all vectors in R³that satisfy the equation x₁ = 3x2, is a subspace.

A: (.)  Let set  'S'  is given by ,        S=x1,x2,x3∈ℝ3  :  x1=3x2 or   S= x1,x2,x3∈ℝ3 :  x1-3x2=0…

Q: Find the closest point to y in the subspace W spanned by u₁ and ₂. y = 19, Un 28 国 26 20 -2 -26…

A:

Q: Prove that if U and V are subspaces of R" and U <V then V+<U+.

A:

Q: 3. Determine whether the set W = {(0,x2, 3, x4) : where x2 and x4 are real numbers} is a subspace of…

A: Subspace of a vector space

Q: 5. Let U = {(u₁, U2, U3) E R³|u₁ + u2 - u3 = 0} and V = {(V1, V2, V3) E R³v₁ - 3v2 +5v3 = 0}. (a)…

A: (a)

Q: 8. Let W denote the subspace of R5 consisting of all the vectors having coordinates that sum to…

A: Consider the provided question,Write the given vectors into a matrix as below,

Q: 2t 6. Let H be the set of all vectors Show that H is a subspace of R³. What is the -t dimension of…

A:

Q: 3. Is the set of all vectors (x, y) in R2 with the usual addition and scalar multiplication, a…

A:

Q: 6. Is the set of points {(x₁, x2 + 1) |x₁, x₂ € R} a subspace of R²? 7. Is the set of points…

A: Subspace of the vector space

Q: 3. Suppose W (a) (b) a = = {( 2 ): 9₂ Prove that W is a subspace of M2x2 (R). : a, b, c ER CER}.…

A: