Suppose U and W are finite-dimensional subspaces of an inner product space V. Prove that Pʊ Pw = 0 if and only if (u, w) = 0 for all u EU and all w Є W. Hint: By the equation PuPw = 0, we mean that for any vector v Є V, PUPwv = 0.

Q: Let 01, 02, 03 be three nonzero vectors in a vector space V. Suppose that V1, V2, V3 are orthogonal…

A:

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 two bases B= (b, b2) and C= (c, c2) for a vector space V such that b, =e, - 6cz and by =…

A: Given two bases  and  for a vector space V such that and . Also, given that and we have to find .…

Q: (2. Let V be a vector space, and let v E V. Show that {v} is linearly dependent if and only if v =…

A:

Q: Show if W is a subspace of the vector space X, then W is convex.

A: Definition used: A set K in a vector space V is called convex if and only if  x, y belongs to K,…

Q: Let V be a vector space over R. Prove that if a E R and T E V satisfy au = 0, then a = 0 or = 0.…

A:

Q: If A is m X n, prove that every vector in null(A) is orthogonal to every vector in row(A).

A: Let A be an m×n matrix. We want to prove that every vector in the null space of A is orthogonal to…

Q: Suppose U and W are subspaces of a vector space V. Prove that if U C W, then U + W = W.

A:

Q: Let U1, U2 and W be the subspaces of a vector space V. Prove or give a counter example if V = U1 ⊕ W…

A: U1,U2 and W be the subspaces of a vector space V.

Q: 5 Suppose that u = 1 2 and v= 0 are vectors in R³. Determine the smallest 2 --- 2 subspace H of R³…

A:

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

A:

Q: Consider the vector space V = P over the field of real numbers F = R. Is the vector 4 – x – x 2 in…

A:

Q: Suppose u, v, and w are vectors in R³ and W = span{u,v,w}. Let x be a vector in R³ that is…

A: We are given that x is orthogonal to each of the vectors u, v, and w. Thus, x is orthogonal to any…

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: