Linear Algebra

118 Chapter Two. Vector Spaces1.31 Theorem 1.12 shows that, with respect to a basis, every linear combination isunique. If a subset is not a basis, can linear combinations be not unique? If so,must they be?̌ 1.32 A square matrix is symmetric if for all indices i and j, entry i, j equals entryj, i.(a) Find a basis for the vector space of symmetric 2×2 matrices.(b) Find a basis for the space of symmetric 3×3 matrices.(c) Find a basis for the space of symmetric n×n matrices.̌ 1.33 We can show that every basis for R 3 contains the same number of vectors.(a) Show that no linearly independent subset of R 3 contains more than threevectors.(b) Show that no spanning subset of R 3 contains fewer than three vectors. (Hint.Recall how to calculate the span of a set and show that this method, when appliedto two vectors, cannot yield all of R 3 .)1.34 One of the exercises in the Subspaces subsection shows that the set{( xyz)∣∣x + y + z = 1}is a vector space under these operations.( ) ( ) ( )x1 x2 x1 + x 2 − 1y 1 + y 2 = y 1 + y 2z 1 z 2 z 1 + z 2Find a basis.r( xyz)=( )rx − r + 1ryrzIII.2 DimensionIn the prior subsection we defined the basis of a vector space, and we saw thata space can have many different bases. For example, following the definition ofa basis, we saw three different bases for R 2 . So we cannot talk about “the” basisfor a vector space. True, some vector spaces have bases that strike us as morenatural than others, for instance, R 2 ’s basis E 2 or R 3 ’s basis E 3 or P 2 ’s basis〈1, x, x 2 〉. But, for example in the space {a 2 x 2 + a 1 x + a 0∣ ∣ 2a 2 − a 0 = a 1 }, noparticular basis leaps out at us as the most natural one. We cannot, in general,associate with a space any single basis that best describes that space.We can, however, find something about the bases that is uniquely associatedwith the space. This subsection shows that any two bases for a space have thesame number of elements. So, with each space we can associate a number, thenumber of vectors in any of its bases.This brings us back to when we considered the two things that could bemeant by the term ‘minimal spanning set’. At that point we defined ‘minimal’as linearly independent, but we noted that another reasonable interpretation ofthe term is that a spanning set is ‘minimal’ when it has the fewest number ofelements of any set with the same span. At the end of this subsection, after we