a. Let V be a subspace of R\". Let m be the largest number of linearly independe

Question

a. Let V be a subspace of R". Let m be the largest number of linearly independent vectors we can find in V. (Note that m n, by Theorem 3.2.8.) Choose linearly independent vectors vi, v2, , Vm in V Show that the vectors v1, v2, , Vm span V and are therefore a basis of V. This exercise shows that any subspace of R" has a basis. If you are puzzled, think first about the special case when V is a plane in R3. What is m in this case? b. Show that any subspace V of R" can be represented as the image of a matrix

Explanation / Answer

b.

Let, W be the subspace of V

Now consider, S = subset of basis set which is contained in W

This must be non empty otherwise we can find a vector in W which is linearly independent from basis set and hence get a larger basis which is not possible

Now , S must be a maximal linearly independent set in W otherwise again we can construct a basis set larger than S and that is not possible

Hence, S is a basis for W

Let, v1,...,vn be basis for Rn

And, vn1,.....,vnk be the k basis vectors from S present in W

Now consider the matrix A whose all column are 0 except : n1,....nk th columns

And in ni th column all entries except in ni the row is equal to 1.

This is the required matrix which projects V to W

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