a)Find bases for the following subspaces of F^5 V = {(c1, c2, c3, c4, c5) | c1 − c3 − c4 = 0} W = {(c1, c2, c3, c4, c5) | c2 = c3 = c4 and c1 + c5 = 0} b)What are their dimensions?

Advanced Engineering Mathematics
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Chapter2: Second-order Linear Odes
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a)Find bases for the following subspaces of F^5

V = {(c1, c2, c3, c4, c5) | c1 − c3 − c4 = 0}

W = {(c1, c2, c3, c4, c5) | c2 = c3 = c4 and c1 + c5 = 0}

b)What are their dimensions?

Expert Solution
Step 1

Given that

V , W are two subspaces of F^5.

We know that

A basis for a subspace S of Rn is a set of vectors in S that is linearly independent and is maximal with this property (that is, adding any other vector in S to this subset makes the resulting set linearly dependent).

The dimension of a nonzero subspace H, denoted by dim H, is the number of vectors in any basis for H. The dimension of the zero space is zero. Definition. Given an m × n matrix A, the rank of A is the maximum number of linearly independent column vectors in A.

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