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Chapter 6: Problem 20

Let \(B=\left\\{\mathbf{v}_{1}, \ldots, \mathbf{v}_{n}\right\\}\) be a maximal independent set in a vector space \(V\). That is, no set of more than \(n\) vectors \(S\) is independent. Show that \(B\) is a basis of \(V\)

Short Answer

Expert verified
\(B\) is a basis of \(V\) because it is a maximal independent set and spans \(V\).

Step by step solution

01

Define a Basis

A basis of a vector space \(V\) is a set of vectors that is both linearly independent and spans \(V\). This implies that every vector in \(V\) can be written as a linear combination of the vectors in the basis.
02

Prove Linear Independence of B

Since \(B\) is given as a maximal independent set, no additional vector can be added to \(B\) without losing its independence property. This means that \(B\) is a linearly independent set of vectors.
03

Show that B Spans V

Suppose \(B\) does not span \(V\). Then there exists a vector \(\mathbf{w} \in V\) that cannot be expressed as a linear combination of the vectors in \(B\). If we add \(\mathbf{w}\) to \(B\), the set \(B \cup \{\mathbf{w}\}\) must still be independent. This contradicts the maximal independence of \(B\), hence \(B\) must span \(V\).
04

Conclusion

Since \(B\) is both linearly independent and spans \(V\), by the definition of a basis, \(B\) is a basis of \(V\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Basis of a Vector Space
Understanding what a basis is in a vector space is crucial. A basis is a set of vectors within a vector space that has two key properties: it is linearly independent, and it spans the entire vector space. This means any vector in the vector space can be represented as a unique combination of the basis vectors.
A simple way to test if a set of vectors forms a basis is by checking these two properties. If the set passes both, it indeed is a basis.
In practical terms, finding a basis helps us simplify calculations in various fields like physics and engineering by reducing complex systems into manageable parts.
Linear Independence
Linear independence is a condition where no vector in a set can be written as a linear combination of the others. This means there are no dependent relationships among the vectors in the set.
To determine if a set of vectors \{\mathbf{v}_{1}, \mathbf{v}_{2}, \ldots, \mathbf{v}_{n}\}\ is linearly independent, consider the combination \(c_1\mathbf{v}_{1} + c_2\mathbf{v}_{2} + \ldots + c_n\mathbf{v}_{n} = 0\).
If the only solution is \(c_1 = c_2 = \ldots = c_n = 0\), then the vectors are independent. Otherwise, they are dependent.
This independence ensures that no vector "repeats" or is overly similar within the set, which is critical for forming a basis.
Spanning Set
A spanning set is a collection of vectors that can represent every vector in the vector space through linear combinations. But not every spanning set is efficient.
Here's why: imagine having too many vectors, some of which might be unnecessary because they result in redundancy. The beauty of a basis is that it is a minimal spanning set, containing only the essential vectors to capture the whole space.
When you are determining whether a set spans a vector space, verify if every vector in the space can be decomposed into a linear combination of the set's vectors.
If any vector is left out, the set fails to span the entire space.

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Most popular questions from this chapter

a. Let \(V=\left\\{\left(x^{2}+x+1\right) p(x) \mid p(x)\right.\) in \(\left.\mathbf{P}_{2}\right\\} .\) Show that \(V\) is a subspace of \(\mathbf{P}_{4}\) and find \(\operatorname{dim} V\). [Hint: If \(f(x) g(x)=0\) in \(\mathbf{P},\) then \(f(x)=0\) or \(g(x)=0 .]\) b. Repeat with \(V=\left\\{\left(x^{2}-x\right) p(x) \mid p(x)\right.\) in \(\left.\mathbf{P}_{3}\right\\},\) a subset of \(\mathbf{P}_{5}\) c. Generalize.

If \(U\) and \(W\) are subspaces of \(V,\) define their intersection \(U \cap W\) as follows: \(U \cap W=\\{\mathbf{v} \mid \mathbf{v}\) is in both \(U\) and \(W\\}\) a. Show that \(U \cap W\) is a subspace contained in \(U\) and \(W\). b. Show that \(U \cap W=\\{\mathbf{0}\\}\) if and only if \(\\{\mathbf{u}, \mathbf{w}\\}\) is independent for any nonzero vectors \(\mathbf{u}\) in \(U\) and \(\mathbf{w}\) in \(W\). c. If \(B\) and \(D\) are bases of \(U\) and \(W,\) and if \(U \cap W=\) \(\\{\boldsymbol{0}\\},\) show that \(B \cup D=\\{\mathbf{v} \mid \mathbf{v}\) is in \(B\) or \(D\\}\) is independent.

Are the following sets vector spaces with the indicated operations? If not, why not? a. The set \(V\) of nonnegative real numbers; ordinary addition and scalar multiplication. b. The set \(V\) of all polynomials of degree \(\geq 3\), together with 0 ; operations of \(\mathbf{P}\). c. The set of all polynomials of degree \(\leq 3\); operations of \(\mathbf{P}\). d. The set \(\left\\{1, x, x^{2}, \ldots\right\\} ;\) operations of \(\mathbf{P}\). e. The set \(V\) of all \(2 \times 2\) matrices of the form \(\left[\begin{array}{ll}a & b \\ 0 & c\end{array}\right] ;\) operations of \(\mathbf{M}_{22}\) f. The set \(V\) of \(2 \times 2\) matrices with equal column sums; operations of \(\mathbf{M}_{22}\). g. The set \(V\) of \(2 \times 2\) matrices with zero determinant; usual matrix operations. h. The set \(V\) of real numbers; usual operations. i. The set \(V\) of complex numbers; usual addition and multiplication by a real number. j. The set \(V\) of all ordered pairs \((x, y)\) with the addition of \(\mathbb{R}^{2},\) but using scalar multiplication \(a(x, y)=(a x,-a y)\) \(\mathrm{k}\). The set \(V\) of all ordered pairs \((x, y)\) with the addition of \(\mathbb{R}^{2}\), but using scalar multiplication \(a(x, y)=(x, y)\) for all \(a\) in \(\mathbb{R}\) 1\. The set \(V\) of all functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) with pointwise addition, but scalar multiplication defined by \((a f)(x)=f(a x)\) \(\mathrm{m}\). The set \(V\) of all \(2 \times 2\) matrices whose entries sum to \(0 ;\) operations of \(\mathbf{M}_{22}\). n. The set \(V\) of all \(2 \times 2\) matrices with the addition of \(\mathbf{M}_{22}\) but scalar multiplication \(*\) defined by \(a * X=a X^{T}\).

If \(z\) is a complex number, show that \(\left\\{z, z^{2}\right\\}\) is independent if and only if \(z\) is not real.

\(\begin{array}{ll}\text { } & \text { By calculating }(1+1)(\mathbf{v}+\mathbf{w}) \text { in two }\end{array}\) ways (using axioms \(\mathrm{S} 2\) and \(\mathrm{S} 3\) ), show that axiom \(\mathrm{A} 2\) follows from the other axioms.
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