# 16.2: Review Problems

1.

Consider an arbitrary matrix \(M:\Re^{m}\to \Re^{n}.\)

a) Argue that \(Mx=0\) if only if \(x\) is perpendicular to all columns of \(M^{T}\).

b) Argue that \(Mx=0\) if only if \(x\) is perpendicular to all of the linear combinations of the columns of \(M^{T}\).

c) Argue that \(\ker M\) is perpendicular to \({\rm ran} \,M^{T}\).

d) Argue further \(\Re^{m}=\ker M \oplus {\rm ran}\, M^{T}\).

e) Argue analogously that \(\Re^{n}=\ker M^{T} \oplus {\rm ran}\, M\).

The equations in the last two parts describe how a linear transformation \(M:\Re^{m}\to \Re^{n}\) determines orthogonal decompositions of both it's domain and target. This result sometimes goes by the humble name \(\textit{The Fundamental Theorem of Linear Algebra}\).

2. Let \(L \colon V\rightarrow W\) be a linear transformation. Show that \(\ker L=\{0_{V}\}\) if and only if \(L\) is one-to-one:

a) \(\textit{(Trivial kernel \(\Rightarrow\) injective.)}\) Suppose that \(\ker L=\{0_V\}\). Show that \(L\) is one-to-one. Think about methods of proof--does a proof by contradiction, a proof by induction, or a direct proof seem most appropriate?

b) \(\textit{(Injective \(\Rightarrow\) trivial kernel.)}\) Now suppose that \(L\) is one-to-one. Show that \(\ker L=\{0_{V}\}\). That is, show that \(0_{V}\) is in \(\ker L\), and then show that there are no other vectors in \(\ker L\).

3. Let \(\{v_{1}, \ldots, v_{n} \}\) be a basis for \(V\). Carefully explain why

\[

L(V)= span \{Lv_{1}, \ldots, Lv_{n} \}.

\]

4. Suppose \(L \colon \Re^{4}\rightarrow \Re^{3}\) whose matrix \(M\) in the standard basis is row equivalent to the following matrix:

\[

\begin{pmatrix}

1&0&0&\!-1\\

0&1&0&1\\

0&0&1&1\\

\end{pmatrix}={\rm RREF}(M)\sim M.

\]

a) \(\textit{Explain}\) why the first three columns of the original matrix \(M\) form a basis for \(L(\Re^{4})\).

b) \(\textit{Find and describe}\) an algorithm (\(\textit{i.e.}\), a general procedure) for computing a basis for \(L(\Re^{n})\) when \(L \colon \Re^{n}\rightarrow \Re^{m}\).

c) \(\textit{Use}\) your algorithm to find a basis for \(L(\Re^{4})\) when \(L \colon \Re^{4} \to \Re^{3}\) is the linear transformation whose matrix \(M\) in the standard basis is

\[

\begin{pmatrix}

2&1&1&4\\

0&1&0&5\\

4&1&1&6\\

\end{pmatrix}.

\]

5. Claim:

If \(\{v_{1}, \ldots, v_{n} \}\) is a basis for \(\ker L\), where \(L \colon V\rightarrow W\), then it is always possible to extend this set to a basis for \(V\).

Choose some simple yet non-trivial linear transformations with non-trivial kernels and verify the above claim for those transformations.

6. Let \(P_{n}(x)\) be the space of polynomials in \(x\) of degree less than or equal to \(n\), and consider the derivative operator $$\frac{d}{dx}:P_{n}(x)\to P_{n}(x)\, .$$

Find the dimension of the kernel and image of this operator. What happens if the target space is changed to \(P_{n-1}(x)\) or \(P_{n+1}(x)\)?

Now consider \(P_{2}(x,y)\), the space of polynomials of degree two or less in \(x\) and \(y\). (Recall how degree is counted; \(xy\) is degree two, \(y\) is degree one and \(x^{2}y\) is degree three, for example.) Let

$$L:=\frac{\partial}{\partial x}+\frac{\partial}{\partial y}:P_{2}(x,y) \to P_{2}(x,y).$$

(For example, \(L(xy)=\frac{\partial}{\partial x}(xy)+\frac{\partial}{\partial y}(xy) = y+x\).) Find a basis for the kernel of \(L\). Verify the dimension formula in this case.

7.

Lets demonstrate some ways the dimension formula can break down if a vector space is infinite dimensional:

a) Let \(\mathbb{R}[x]\) be the vector space of all polynomials in the variable \(x\) with real coefficients. Let \(D = \frac{d}{dx}\) be the usual derivative operator. Show that the range of \(D\) is \(\mathbb{R}[x]\). What is \(\ker D\)? \(\textit{Hint: Use the basis \({ x^{n} \mid n \in \mathbb{N} }.\)}\)

b) Let \(L \colon \mathbb{R}[x] \rightarrow \mathbb{R}[x]\) be the linear map $$L(p(x)) = xp(x)\, .$$ What is the kernel and range of \(M\)?

c) Let \(V\) be an infinite dimensional vector space and \(L \colon V \rightarrow V\) be a linear operator. Suppose that \(\dim \ker L < \infty\), show that \(\dim L(V)\) is infinite. Also show that when \(\dim L(V) < \infty\) that \(\dim \ker L\) is infinite.

8. This question will answer the question, "If I choose a bit vector \(\textit{at random}\), what is the probability that it lies in the span of some other vectors?''

[\(i.\)] Given a collection \(S\) of \(k\) bit vectors in \(B^{3}\), consider the bit matrix \(M\) whose columns are the vectors in \(S\). Show that \(S\) is linearly independent if and only if the kernel of \(M\) is trivial, namely the set \({\rm ker}M=\{v\in B^{3}| \, Mv=0\}\) contains only the zero vector.

[\(ii.\)] Give some method for choosing a random bit vector \(v\) in \(B^{3}\). Suppose \(S\) is a collection of \(2\) linearly independent bit vectors in \(B^{3}\). How can we tell whether \(S\cup \{v\}\) is linearly independent? Do you think it is likely or unlikely that \(S\cup \{v\}\) is linearly independent? Explain your reasoning.

[\(iii.\)] If \(P\) is the characteristic polynomial of a \(3 \times 3\) bit matrix, what must the degree of \(P\) be? Given that each coefficient must be either 0 or 1, how many possibilities are there for \(P\)? How many of these possible characteristic polynomials have \(0\) as a root? If \(M\) is a \(3 \times 3\) bit matrix chosen at random, what is the probability that it has 0 as an eigenvalue? (Assume that you are choosing a random matrix \(M\) in such a way as to make each characteristic polynomial equally likely.) What is the probability that the columns of \(M\) form a basis for \(B^{3}\)? \((\textit{Hint: what is the relationship between the kernel of \(M\) and its eigenvalues?})\)

[Note:] We could ask the same question for real vectors: If I choose a real vector at random, what is the probability that it lies in the span of some other vectors? In fact, once we write down a reasonable way of choosing a random real vector, if I choose a real vector in \(\Re^{n}\) at random, the probability that it lies in the span of \(n-1\) other real vectors is zero!

### Contributor

- David Cherney, Tom Denton, and Andrew Waldron (UC Davis)