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Chapter 3: Problem 7

Find a basis for the subspace \(S\) of \(\mathbb{R}^{4}\) consisting of all vectors of the form \((a+b, a-b+2 c, b, c)^{T}\) where \(a, b,\) and \(c\) are all real numbers. What is the dimension of \(S ?\)

Short Answer

Expert verified
A basis for the subspace S is given by \(\{(1, 1, 0, 0)^T, (1, -1, 1, 0)^T, (0, 2, 0, 1)^T\}\), and the dimension of S is 3.

Step by step solution

01

Represent the Vector in Terms of its Components

\ Given a vector in S as \((a+b, a-b+2c, b, c)^T\), we can represent it in terms of its components a, b, and c: \[v = a(1, 1, 0, 0)^T + b(1, -1, 1, 0)^T + c(0, 2, 0, 1)^T.\]
02

Identify the Basis Vectors

\ The given vector can be generated by linear combinations of the vectors: \[(1, 1, 0, 0)^T, (1, -1, 1, 0)^T, (0, 2, 0, 1)^T.\] These vectors are linearly independent, as it is not possible to express one as a linear combination of the others. Therefore, they form a basis for the subspace S.
03

Find the Dimension of S

\ Since the basis for the subspace S consists of 3 linearly independent vectors, the dimension of S is 3. In summary, a basis for the subspace S is given by: \[\{(1, 1, 0, 0)^T, (1, -1, 1, 0)^T, (0, 2, 0, 1)^T\},\] and the dimension of S is 3.

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Most popular questions from this chapter

In each of the following, find the dimension of the subspace of \(P_{3}\) spanned by the given vectors: (a) \(x, x-1, x^{2}+1\) (b) \(x, x-1, x^{2}+1, x^{2}-1\) (c) \(x^{2}, x^{2}-x-1, x+1\) (d) \(2 x, x-2\)

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