Answer all the questions. Please provide a small explanation too. MA 2033 Linear

Question

Answer all the questions. Please provide a small explanation too.

MA 2033 Linear Algebra - Mid Semester Examination November 2015 Time: I hour Answer ALL Questions. Index No: Field: Cirele the Correct Answer Wa,be)eR',a+c is a vector space TRUE/ FALSH 2. Ia set spas a finite dimensional vestor spase / and if 7 is a set of more than n vectors in V, then T is linearly dependent. TRUE/FALSE 3· Dim (Rh)-n, where P, is the set of all polynomials of degree less than or equal to n. TRUE /FALSE 4. The empty set is a subspace of every vector spoce TRUEYFALSE s. Let Ube a subspace of an n-dimensional vecsor space. Then dim(U)sn TRUE /FALSE 6. The span of the empty set is TRUE / FALSE 7. A set consisting of a single non-zero vector is linearly independent. TRUE/ FALSE 8. IFP Q R and S are subspaces of a vector space V, then PnenR n S is a subspace of V. TRUE /FALSE Write answers to the following questions in the space provided. 9, and W are subspaces of a vector space ). Define the set t 10. A set B = { {y,,y,, ,y,h is called a basis for the vector space Vit. 11. Find a basis for the vector space V = {(x, y, z, x+p-z.x-y-r) and hence fin the dimension of Continued..

Explanation / Answer

1. TRUE

2. TRUE

If T is a set of more than n vectors in V, then T is linearly dependent because there will exist a Scalar multiple 'k' which is non-zero. So it has to be a linear function of other vectors.

3. TRUE

The set {1,x,x2...,xk}{1,x,x2...,xk} form a basis of the vector space of all polynomials of degree kk over some field. Every polynomial will be in some linear combination of these vectors. Also it is not difficult to show that the above set is linear independent. So dimension of the vector space is k+1k+1. Your vector space has infinite polynomials but every polynomial has degree kk and so is in the linear span of the set {1,x,x2...,xk}{1,x,x2...,xk}.

4. TRUE

we can look at it in two ways:

a. The set A is a subset of the set B if and only if every element of A is also an element of B. If A is the empty set then A has no elements and so all of its elements (there are none) belong to B no matter what set B we are dealing with. That is, the empty set is a subset of every set.

b. Another way of understanding it is to look at intersections. The intersection of two sets is a subset of each of the original sets. So if {} is the empty set and A is any set then {} intersect A is {} which means {} is a subset of A and {} is a subset of {}.

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