How to prove or counter with example the following statements:
(1) If two subspaces are orthogonal, then they are independent.
(2) If two subspaces are independent, then they are orthogonal.
I know that a vector v element of V is orthogonal to a subspace W element V if v is orthogonal to every w element W. Two subspaces W1

Give a demonstration as to why or why not the given objects are vector subspaces of M22
a) all 2 X 2 matrices with integer entries
A vector space is a set that is closed under finite vector addition and scalar multiplication.
It is not a vector space, since V is NOT closed under finite scalar multiplication. For insta

2. Use Theorem 5.2.1 to determine which of the following are subspaces of R3.
Thm 5.2.1: If W is a set of one or more vectors from a vector space V, then W is a subspace of V if and only if the following conditions hold.
(a) If u and v are vectors in W, then u + v is in W.
(b) If k is any scalar and u is any vector in W,

Please help with the following problems.
1. Let u1 = (1,2,1,-1) and u2 = (2,4,2,0). Extend the linearly independent set {u1,u2} to obtain a basis for R4 (reals in 4 dimensions)
2. Let U1,U2 be two subspaces of a finite dimensional vector space V such that U1+U2 = V. Prove that there is a subspace W of U1 such that W (+)

Let V be the space of all functions from R to R. It was stated in the discussion session that this is a vector space over R
Let F be a field, V a vector space over F, and v1,...,vk vectors in V. Prove that the set Span({v1, ..., vk}) is closed under scalar multiplication.
1. Label the following statements as true or false

Let [a,b] be an interval in {see attachment}. Recall that the set of functions {see attachment} is a vector space over {see attachment} with addition (f+g)(x):=f(x)+g(x) and scalar multiplication
a) choose [a,b]=[0,1]. Decide for each of the following subsets if it is a subspace. Justify your answer by giving a proof or a c

Define vectors pace and subspace with examples.
State and prove a necessary and sufficient condition for a subset of vectors to be a subspace.
Show that the intersection and union of two sub spaces are also sub spaces.

Please show all work. Thanks. See attached for proper formatting
Question 1
Let B={v1,...,vn} be a basis of a subspace V of Rnx1. Let x be the nonzero vector x=a1v1+...+anvn for scalars ai. Let C={x, v2,...,vn}.
a) Show that if a1 is not equal to 0 then C is also a basis.
b) Show that if a1 =0 then C is not a basis.