Linear Algebra

Section III. Basis and Dimension 119The main result of this subsection, that all of the bases in a finite-dimensionalvector space have the same number of elements, is the single most importantresult in this book because, as Example 2.11 shows, it describes what vectorspaces and subspaces there can be. We will see more in the next chapter.One immediate consequence brings us back to when we considered the twothings that could be meant by the term ‘minimal spanning set’. At that point wedefined ‘minimal’ as linearly independent but we noted that another reasonableinterpretation of the term is that a spanning set is ‘minimal’ when it has thefewest number of elements of any set with the same span. Now that we haveshown that all bases have the same number of elements, we know that the twosenses of ‘minimal’ are equivalent.ExercisesAssume that all spaces are finite-dimensional unless otherwise stated.̌ 2.15 Find a basis for, and the dimension of, P 2 .2.16 Find a basis for, and the dimension of, the solution set of this system.x 1 − 4x 2 + 3x 3 − x 4 = 02x 1 − 8x 2 + 6x 3 − 2x 4 = 0̌ 2.17 Find a basis for, and the dimension of, M 2×2 , the vector space of 2×2 matrices.2.18 Find the dimension of the vector space of matrices( ) a bsubject to each condition.cd(a) a, b, c, d ∈ R(b) a − b + 2c = 0 and d ∈ R(c) a + b + c = 0, a + b − c = 0, and d ∈ Ř 2.19 Find the dimension of each.(a) The space of cubic polynomials p(x) such that p(7) = 0(b) The space of cubic polynomials p(x) such that p(7) = 0 and p(5) = 0(c) The space of cubic polynomials p(x) such that p(7) = 0, p(5) = 0, and p(3) = 0(d) The space of cubic polynomials p(x) such that p(7) = 0, p(5) = 0, p(3) = 0,and p(1) = 02.20 What is the dimension of the span of the set {cos 2 θ, sin 2 θ, cos 2θ, sin 2θ}? Thisspan is a subspace of the space of all real-valued functions of one real variable.2.21 Find the dimension of C 47 , the vector space of 47-tuples of complex numbers.2.22 What is the dimension of the vector space M 3×5 of 3×5 matrices?̌ 2.23 Show that this is a basis for R 4 .⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞1 1 1 1〈 ⎜0⎟⎝0⎠ , ⎜1⎟⎝0⎠ , ⎜1⎟⎝1⎠ , ⎜1⎟⎝1⎠ 〉0 0 0 1(We can use the results of this subsection to simplify this job.)2.24 Refer to Example 2.11.(a) Sketch a similar subspace diagram for P 2 .(b) Sketch one for M 2×2 .