- Show that the following are subspaces of the set of all functions defined on .
- polynomials of degree ≤ n
- polynomials
- continuous functions
- differentiable functions

- Show that every subspace of a finite dimensional vector space V is the span of some vectors. It was done above but go over it in your own words.
- In ℝ
^{2}define a funny addition by+≡and let scalar multiplication be the usual thing. Would this be a vector space with these operations? - Determine which of the following are subspaces of ℝ
^{m}for some m. a,b are just given numbers in what follows.- For those who recall the cross product,
- For those who recall the dot product,
- . This is known as S
^{⊥}.

- Show that is a subspace and find a basis for it.
- In the subspace of polynomials on , show that the vectorsare linearly independent. Show these vectors are a basis for the vector space of polynomials of degree no more than 3.
- Determine whether the real valued functions defined on ℝ
are linearly independent. Is this a basis for the subspace of polynomials of degree no more than 3? Explain why or why not.

- Determine whether the real valued functions defined on ℝ
are linearly independent. Is this a basis for the subspace of polynomials of degree no more than 3? Explain why or why not.

- Show that the following are each a basis for ℝ
^{3}.- ,,
- ,,
- ,,
- ,,

- Show that each of the following is not a basis for ℝ
^{3}. Explain why they fail to be a basis.- ,,
- ,,
- ,,
- ,
- ,,,

- Suppose B is a subset of the set of complex valued functions, none equal to 0 and defined on Ω and it has the property that if f,g are different, then fg = 0. Show that B must be linearly independent.
- Suppose you have continuous real valued functions defined on ,and these satisfy
Show that these functions must be linearly independent.

- Show that the real valued functions cos,1,cos
^{2}are linearly dependent. - Show that the real valued functions e
^{x}sin,e^{x}cosare linearly independent. - Let the field of scalars be ℚ and let the vector space be all vectors (real numbers) of the
form a + bfor a,b ∈ ℚ. Show that this really is a vector space and find a basis for it.
- Consider the two vectors ,in ℝ
^{2}. Show that these are linearly independent. Now consider,in ℤ_{3}^{2}where the numbers are interpreted as residue classes. Are these vectors linearly independent? If not, give a nontrivial linear combination which is 0. - Is ℂ a vector space with field of scalars ℝ? If so, what is the dimension of this vector space? Give a basis.
- Is ℂ a vector space with field of scalars ℂ? If so, what is the dimension? Give a basis.
- The space of real valued continuous functions on usually denoted as Cis a vector space with field of scalars ℝ. Explain why it is not a finite dimensional vector space.
- Suppose two vector spaces V,W have the same field of scalars F. Show that V ∩ W is a subspace of both V and W.
- If V,W are two sub spaces of a vector space U, define V + W ≡. Show that this is a subspace of U.
- If V,W are two sub spaces of a vector space U, consider V ∪W, the vectors which are in either V or W. Will this be a subspace of U? If so, prove it is the case and if not, give an example which shows that it is not necessarily true.
- Let V,W be vector spaces. A function T : V → W is called a linear transformation if whenever α,β
are scalars and u,v are vectors in V , it follows that
Then ker

≡,Im≡. Show the first of these is a subspace of V and the second is a subspace of W. - ↑In the situation of the above problem, where T is a linear transformation, suppose S is a linearly
independent subset of W. Define T
^{−1}≡. Show that T^{−1}is linearly independent. - ↑In the situation of the above problems, rank is defined as the dimension of Im. Also the nullity of T, denoted as nullis defined as the dimension of ker. In this problem, you will show that if the dimension of V is n, then rank+ null= n.
- Let a basis for kerbe. Let a basis for Imbe. You need to show that r+s = n. Begin with u ∈ V and consider Tu. It is a linear combination ofsay ∑
_{i=1}^{s}a_{i}Tv_{i}. Why? - Next explain why T= 0 . Then explain why there are scalars b
_{j}such that u −∑_{i=1}^{s}a_{i}v_{i}= ∑_{j=1}^{r}b_{j}z_{j}. - Observe that V = span. Why?
- Finally show that is linearly independent. Thus n = r + s.

- Let a basis for ker

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