Now that we have the notions of span and linear independence in place, we simply combine them to define a basis of a vector space. In the spirit of SectionΒ 3.5, a basis of a vector space \(V\) should be understood as a minimal spanning set.
This section includes many theoretical results. There are two in particular that are worth highlighting, especially in regard to computational techniques for abstract vector spaces:
If \(B\) is a basis of \(V\) containing exactly \(n\) elements, then any other basis \(B'\) also contains exactly \(n\) elements. (TheoremΒ 3.7.3)
If \(B\) is a basis for \(V\text{,}\) then every element of \(V\) can be written as a linear combination of elements of \(B\) in a unique way. (TheoremΒ 3.6.7)
The first result allows us to define the dimension of a vector space as the number of elements in any given basis. The second result allows us to take any \(n\)-dimensional vector space \(V\) with chosen basis \(B=\{\boldv_1, \boldv_2, \dots, \boldv_n\}\) and effectively identify vectors \(\boldv\in V\) with the sequence \((c_1,c_2,\dots, c_n)\in \R^n\text{,}\) where
This observation has the following consequence: given any \(n\)-dimensional vector space \(V\text{,}\) no matter how exotic, once we choose a basis \(B\) of \(V\text{,}\) we can reduce any and all linear algebraic questions or computations about \(V\) to a corresponding question in \(\R^n\text{.}\) We will elaborate this idea further in SectionΒ 4.1.
The examples of standard spanning sets in RemarkΒ 3.5.7 are easily seen to be linearly independent, and hence are in fact bases. We list them again here, using the same notation, and refer to these as standard bases for the given spaces.
Let \(V=\{\boldzero\}\text{.}\) The empty \(B=\emptyset=\{ \}\) is a basis for \(V\text{.}\) Note that \(B=\emptyset\) spans \(V\) by definition (DefinitionΒ 3.5.1), and it satisfies the defining implication of linear independence (DefinitionΒ 3.5.9) trivially.
Each verification amounts to showing, using the techniques from SectionΒ 3.5, that the given \(B\) spans the given \(V\) and is linearly independent. We illustrate with (1) and (2), leaving (3) to the reader.
Since neither element of \(B=\{(1,1), (1,-1)\}\) is a scalar multiple of the other, the set is linearly independent. To see that \(B\) spans \(\R^2\) we show that for any \((c,d)\in \R^2\) we have
for some \(a,b\in \R\text{.}\) Indeed we may take \(a=\frac{1}{2}(c+d)\) and \(b=\frac{1}{2}(c-d)\text{.}\) (These formulas were obtained by solving the corresponding system of two equations in the unknowns \(a\) and \(b\text{.}\))
An easy computation shows \(\det A=3\text{,}\) and thus that \(A\) is invertible. We conclude that the system can be solved for \((a,b,c)\) (set \(\boldx=A^{-1}\boldy\)), and thus that \(B\) spans \(P_2\text{.}\)
Our work above now can be used to also show that \(B\) is linearly independent. Replacing the arbitrary polynomial \(dx^2+ex+f\) with the zero polynomial \(0x^2+0x+0\text{,}\) we see that a linear combination
corresponds to a solution \(\boldx=(a,b,c)\) to the matrix equation \(A\boldx=\boldzero\text{.}\) Since \(A\) is invertible, we conclude that \(\boldx=\boldzero\) (TheoremΒ 2.5.27), and thus that \(a=b=c=0\text{.}\) This shows \(B\) is linearly independent.
Indeed let \(S=\{p_1, p_2, \dots, p_r\}\) be a finite set of polynomials, and let \(n\) be the maximal degree of all the polynomials \(p_i\in S\text{.}\) Then \(p_i\in P_n\) for all \(i\text{,}\) in which case \(\Span S\subseteq P_n\text{:}\) i.e., \(\Span S\) is a subspace of the space of polynomials of degree at most \(n\text{.}\) Since \(P_n\subsetneq P\text{,}\) we conclude that \(\Span S\ne P\text{,}\) as claimed.
By TheoremΒ 3.7.16 and its corollaries, we know that if \(V\) has a finite basis, then and subspace of \(V\) also has a finite basis. Let \(X\subseteq \R\) be an interval. Since
Let \(V=\R_{>0}\text{,}\) and let \(S=\{\boldv\}\text{,}\) where \(\boldv=a\) is any positive real number. Prove: \(S\) is a basis if and only if \(a\ne 1\text{.}\)
Suppose \(a=1\text{.}\) Since \(1=\boldzero\in V\text{,}\) we have \(S=\{\boldzero\}\text{.}\) Any set containing the zero vector is linearly dependent (RemarkΒ 3.5.10). Thus \(S\) is not a basis.
Since \(S=\{\boldv\}\) consists of one nonzero element, it is linearly independent (RemarkΒ 3.5.10). It remains only to show that \(S\) spans \(\R_{>0}\text{,}\) which amounts to showing that every \(b\in \R_{>0}\) is a scalar multiple of \(\boldv=a\text{.}\) Since by definition scalar multiplication in \(\R_{>0}\) is defined as \(c\boldv=a^c\text{,}\) this is equivalent to showing that every \(b\in \R_{>0}\) can be written in the form \(b=a^c\text{.}\) This fact is a familiar result from calculus, where you learn that the range (or image) of any exponential function \(f(x)=a^x\) is the set of all positive real numbers.
Proceeding directly from the definition, to show a set \(B\) is a basis of \(V\) we have to do two steps: (i) show \(\Span B= V\text{;}\) (ii) show that \(B\) is linearly independent. The following theorem offers gives rise to a one-step technique for proving \(B\) is a basis: show that every element of \(V\) can be written as a linear combination of elements of \(B\) in a unique way.
Every element \(\boldv\in V\) can be written as a linear combination of elements of \(B\text{,}\) and furthermore this linear combination is unique: i.e., if we have
Suppose \(B\) is a basis. By definition, \(B\) spans \(V\text{,}\) and so every element of \(V\) can be written as a linear combination of elements of \(B\text{.}\) It remains to show that this linear combination is unique in the sense described. This follows from the fact that \(B\) is linearly independent. Indeed, if
Since \(B\) is linearly independent and since the \(\boldv_i\) are distinct, we must have \(c_i-d_j=0\text{,}\) and hence \(c_i=d_i\) for all \(1\leq i\leq r\text{.}\)
If \(B\) satisfies (2), then clearly it spans \(V\text{.}\) The uniqueness of linear combinations of elements of \(B\) now easily implies \(B\) is linearly independent:
Let \(V\) be a vector space. To prove a subset \(B\subseteq V\) is a basis it suffices to show that every \(\boldv\in V\) can be written as a linear combination of elements of \(B\) in a unique way, as specified in TheoremΒ 3.6.7.
has a unique solution \(\boldx=(c_1,c_2,c_3,c_4)\) for any choice of \(\boldy=(a,b,c)\text{.}\) Performing Gaussian elimination on the corresponding augmented matrix yields
Since the third column of \(U\) does not have a leading one, we conclude that the corresponding system has a free variable, and hence that for any given \((a,b,c)\in \R^3\) the equation (3.18) has either no solutions (inconsistent) or infinitely many solutions. In particular, it is not true that there is always a unique solution. Thus \(S\) is not a basis according to the one-step technique.
In fact, our Gaussian elimination analysis tells us exactly how \(S\) fails to be a basis. Since the last column of \(U\) does not have a leading one, the corresponding system is always consistent: i.e., there is always at least one solution \(\boldx=(c_1,c_2,c_3,c_4)\) to (3.18) for each \((a,b,c)\in \R^3\text{.}\) This tells us that \(S\) is a spanning set of \(\R^3\text{.}\) On the other hand, the existence of the free variable tells us that for \((a,b,c)=(0,0,0)=\boldzero\text{,}\) we will have infinitely many choices \(c_1,c_2,c_3,c_4\) satisfying
We conclude that any \(ax+b\in P_1\) can be written as \(c_1p+c_2q\) in a unique way: namely, with \(c_1=a-b\) and \(c_2=-a+2b\text{.}\) Thus \(S\) is a basis.
Besides deciding whether a given set is a basis, we will often want to come up with a basis of a given space on our own. The following βby inspectionβ technique often comes in handy in cases where the elements of the vector space can be described in a simple parametric manner.
Let \(T\) and \(T'\) be linear transformations from \(V\) to \(W\text{.}\) If \(T(\boldu)=T'(\boldu)\) for all \(\boldu\in B\text{,}\) then \(T=T'\text{.}\)
assigning each element of \(\boldu\in B\) to a chosen element \(\boldw_\boldu\in W\) extends uniquely to a linear transformation \(T\colon V\rightarrow W\) satisfying
for all \(\boldu\in B\text{.}\) In more detail, given any \(\boldv\in V\text{,}\) if \(\boldv=c_1\boldu_1+c_2\boldu_2+\cdots +c_r\boldu_r\text{,}\) where \(\boldu_i\in B\) and \(c_i\ne 0\) for all \(1\leq i\leq r\text{,}\) then
Assume \(T\) and \(T'\) are linear transformations from \(V\) to \(W\) satisfying \(T(\boldu)=T'(\boldu)\) for all \(\boldu\in B\text{.}\) Given any \(\boldv\in V\) we can write \(\boldv=c_1\boldu_1+c_2\boldu_2+\cdots +c_r\boldu_r\text{.}\) It follows that
where \(c_i\ne 0\) for all \(1\leq i\leq r\text{,}\) the formula in (3.20) defines a function \(T\colon V\rightarrow W\) in a well-defined manner. Note also that the formula still applies even if some of the coefficients are equal to 0: if \(c_i=0\text{,}\) then \(c_iT(\boldv_i)=\boldzero\text{,}\) and the right-hand side of (3.20) is unchanged. We will use this fact below.
We now show that \(T\) is linear. Given \(\boldv, \boldv'\in V\) we can find a common collection of elements \(\boldu_1,\boldu_2,\dots, \boldu_r\in B\) for which
for some \(c_i, d_i\in \R\text{.}\) We can no longer assume that \(c_i\ne 0\) and \(d_i\ne 0\) for all \(1\leq i\leq r\text{,}\) but as observed above we still have
A linear transformation \(T\colon V\rightarrow W\) is completely determined by its behavior on a basis \(B\subseteq V\text{.}\) Once we know the images \(T(\boldu)\) for all \(\boldu\in B\text{,}\) the image \(T(\boldv)\) for any other \(\boldv\in V\) is then completely determined. Put another way, if two linear transformations out of \(V\)agree on the elements of a basis \(B\subseteq V\text{,}\) then they agree for all elements of \(V\text{.}\)
Once we have a basis \(B\subseteq V\) on hand, it is easy to construct linear transformations \(T\colon V\rightarrow W\text{:}\) simply choose images \(T(\boldu)\in W\) for all \(\boldu\in B\) in any manner you like, and then define \(T(\boldv)\) for any element \(\boldv\in V\) using (3.20).
Let \(r_0\colon \R^2\rightarrow\R^2\) be reflection across the \(x\)-axis, and let \(r_{\pi/2}\colon \R^2\rightarrow \R^2\) be reflection across the \(y\)-axis. (See ExerciseΒ 20.) Use an argument in the spirit of statement (i) from RemarkΒ 3.6.16 to show that
Since \(r_0\) and \(r_{\pi/2}\) are both linear transformations (ExerciseΒ 20), so is the composition \(T=r_{\pi/2}\circ r_{0}\text{.}\) We wish to show \(T=\rho_{\pi}\text{.}\) Since \(\rho_{\pi}\) is also a linear transformation, it suffices by TheoremΒ 3.6.15 to show that \(T\) and \(\rho_\pi\) agree on a basis of \(\R^2\text{.}\) Take the standard basis \(B=\{(1,0), (0,1)\}\text{.}\) Compute:
As a corollary to TheoremΒ 3.6.15 we can at last complete the partial description of linear transformations of the form \(T\colon \R^n\rightarrow \R^m\) given in TheoremΒ 3.2.9.
Given any linear transformation \(T\colon \R^n\rightarrow \R^m\) there is a unique \(m\times n\) matrix \(A\) such that \(T=T_A\text{.}\) In fact we have
where \(B=\{\bolde_1, \bolde_2, \dots, \bolde_n\}\) is the standard basis of \(\R^n\text{.}\) As a result, in the special case where the domain and codomain are both spaces of tuples, all linear transformations are matrix transformations.
In other words, the \(j\)-th column of \(A\) is \(T(\bolde_j)\text{,}\) considered as an \(m\times 1\) column vector. The corresponding matrix transformation \(T_A\colon \R^n\rightarrow \R^m\) is linear by TheoremΒ 3.2.9. Since \(T\) is linear by assumption, TheoremΒ 3.6.15 applies: to show \(T=T_A\) we need only show that \(T(\bolde_j)=T_A(\bolde_j)\) for all \(1\leq j\leq n\text{.}\) We have
\begin{align*}
T_A(\bolde_j) \amp =A\bolde_j \amp (\knowl{./knowl/xref/d_matrix_transform.html}{\text{Definition 3.2.8}})\\
\amp=(j\text{-th column of } A) \amp (\knowl{./knowl/xref/th_column_method.html}{\text{Theorem 2.1.24}}) \\
\amp = T(\bolde_j) \amp (\text{def. of } A)\text{.}
\end{align*}
Besides rounding out our theoretical discussion of linear transformations from \(\R^n\) to \(\R^m\text{,}\) computationally CorollaryΒ 3.6.18 provides a recipe for computing a βmatrix formulaβ for a linear transformation \(T\colon \R^n\colon \rightarrow \R^m\text{.}\) In other words, it tells us how to build the \(A\text{,}\) column by column, such that \(T\boldx=A\boldx\) for all \(\boldx\in R^n\text{.}\) For reasons that will be made more clear in SectionΒ 4.2, we will call \(A\) the standard matrix of \(T\text{.}\)
Definition3.6.19.Standard matrix of linear \(T\colon \R^n\rightarrow \R^m\).
Let \(T\colon \R^n\rightarrow \R^m\) be a linear transformation. The standard matrix of \(T\) is the unique \(m\times n\) matrix \(A\) satisfying \(T=T_A\text{.}\) Equivalently, \(A\) is the unique matrix satisfying
Fix an angle \(\alpha\text{.}\) Taking for granted that the rotation operation \(\rho_\alpha\colon\R^2\rightarrow\R^2\) is a linear transformation, re-derive the matrix formula for \(\rho_\alpha\colon \R^2\rightarrow\R^2\text{:}\) i.e., compute \(A\text{,}\) the standard matrix of \(\rho_\alpha\text{.}\)
since \((1,0)\) gets rotated by \(\rho_\alpha\) to \((\cos\alpha, \sin\alpha)\text{,}\) and \((0,1)\) gets rotated to \((-\sin\alpha, \cos\alpha)\text{.}\)
Find a basis \(\lbrace p(x), q(x) \rbrace\) for the vector space \(\lbrace f(x)\in{\mathbb P}_3[x] \mid f'(-2)=f(1) \rbrace\) where \({\mathbb P}_3[x]\) is the vector space of polynomials in \(x\) with degree less than 3.
Find a basis \(\lbrace p(x), q(x) \rbrace\) for the kernel of the linear transformation \(L:{\mathbb P}_3[x]\to {\mathbb R}\) defined by \(L(f(x)) = f'(-5)-f(1)\) where \({\mathbb P}_3[x]\) is the vector space of polynomials in \(x\) with degree less than 3.
A square matrix is half-magic if the sum of the numbers in each row and column is the same. Find a basis \(B\) for the vector space of \(2\times 2\) half-magic squares.
For each vector space \(V\) and subset \(S\text{,}\) use the one-step technique (ProcedureΒ 3.6.8) to decide whether \(S\) is a basis for \(V\text{.}\)
For each given \(V\) and subspace \(W\subseteq V\text{,}\) provide a basis for \(W\) using the βby inspectionβ technique ProcedureΒ 3.6.13. In more detail:
Let \(S=\{\boldv_1, \boldv_2, \dots, \boldv_k\}\) be a set of \(k\) distinct elements of \(\R^n\text{,}\) let \(A\) be an invertible \(n\times n\) matrix, and let \(S'=\{A\boldv_1, A\boldv_2,\dots, A\boldv_k\}\text{.}\) Prove that \(S\) is a basis of \(\R^n\) if and only if \(S'\) is a basis of \(\R^n\) as follows.
Prove that if \(\{\boldv_1, \boldv_2, \dots, \boldv_k\}\) is a basis of \(\R^n\text{,}\) then \(\{A\boldv_1, A\boldv_2,\dots, A\boldv_k\}\) is also a basis for any invertible \(n\times n\) matrix \(A\text{.}\)
Now prove that if \(\{A\boldv_1, A\boldv_2,\dots, A\boldv_k\}\) is a basis of \(\R^n\) for the invertible matrix \(A\text{,}\) then \(\{\boldv_1, \boldv_2, \dots, \boldv_k\}\) is a basis of \(\R^n\text{.}\)
Hint: in (b) you showed that if you start with any basis and apply any invertible matrix to the elements of this basis, then you end up with another basis; think of a useful choice of matrix for the present situation.
Let \(V=M_{nn}\text{.}\) For each of the following subspaces \(W\subset M_{nn}\text{,}\) give a basis \(B\) of \(W\text{.}\) You must explicitly describe the elements of your basis as linear combinations of the elements \(E_{ij}\) of the standard basis for \(M_{nn}\text{.}\) No justification needed, as long as your proposed basis is simple enough.
The set \(B=\{\boldv_1=(1,-1), \boldv_2=(1,1)\}\) is a basis of \(\R^2\text{.}\) Suppose the linear transformation \(T\colon \R^2\rightarrow \R^3\) satisfies
Suppose \(T\colon V\rightarrow V\) is a linear transformation, and \(B\) is a basis of \(V\) for which \(T(\boldv)=\boldzero\) for all \(\boldv\in B\text{.}\) Show that \(T=0_{V,W}\text{:}\) i.e., \(T\) is the zero transformation from \(V\) to \(W\text{.}\)
Suppose \(T\colon V\rightarrow V\) is a linear transformation, and \(B\) is a basis of \(V\) for which \(T(\boldv)=\boldv\) for all \(\boldv\in B\text{.}\) Show that \(T=\id_V\text{:}\) i.e., \(T\) is the identity transformation of \(V\text{.}\)
Let \(T\colon \R^n\rightarrow \R^n\) be a linear transformation. Assume there is a basis \(B\) of \(\R^n\) and a constant \(c\in \R\) such that \(T(\boldv)=c\boldv\) for all \(\boldv\in B\text{.}\) Prove: \(T=T_{A}\text{,}\) where
For each linear transformation \(T\colon \R^n\rightarrow \R^m\) and \(\boldx\in \R^n\) : (a) compute the standard matrix \(A\) of \(T\) using CorollaryΒ 3.6.18; (b) compute \(T(\boldx)\) using \(A\text{.}\) You may take for granted that the given \(T\) is linear.