Section A

• 1. Let $$U,W\leq \R ^3$$ be given by $$U=\Span {(1,1,1), (1,2,1)}$$, $$W=\Span {(1,0,0),(0,0,1)}$$.

Write down bases for $$U\cap W$$ and $$U+W$$. [4]

• 2. Let $$\phi \in L(V)$$ be a linear operator on a finite-dimensional vector space $$V$$ such that

\begin{equation*} \ker \phi \cap \im \phi =\set {0}. \end{equation*}

Prove that $$\ker \phi \oplus \im \phi =V$$. [4]

• 3. Find the minimal polynomial of $$A\in M_3(\R )$$ given by

\begin{equation*} A= \begin{pmatrix} 0&1&1\\0&0&2\\1&0&0 \end {pmatrix}. \end{equation*}

[4]

• 4. Let $$\phi \in L(V)$$ be a linear operator on a complex vector space $$V$$ with characteristic polynomial $$(x-3)^{4}(x-2)^{2}$$ and minimal polynomial $$(x-3)^{2}(x-2)$$.

What are the possible Jordan normal forms of $$\phi$$? [4]

• 5. Let $$E\leq (\R ^{3})^{*}$$ be spanned by $$\alpha$$ given by

\begin{equation*} \alpha (x)=x_1+2x_2-x_3. \end{equation*}

Write down a basis of $$\sol E$$. [4]

• 6. Define the rank and signature of a symmetric bilinear form $$B$$ on a real, finite-dimensional vector space.

State Sylvester’s Law of Inertia. [4]

Section B

• 7. Let $$V$$ be a vector space over a field $$\F$$ and $$\phi \in L(V)$$ a linear operator on $$V$$.

• (a) Suppose that $$V=V_1\oplus V_2$$ with each $$V_i$$ $$\phi$$-invariant. Show that

\begin{equation*} \ker \phi =\ker \phi \restr {V_1}\oplus \ker \phi \restr {V_2}. \end{equation*}

[6]

• (b) Let $$U\leq V$$ be $$\phi$$-invariant and let $$q:V\to V/U$$ be the quotient map.

• (i) Show that there is a well-defined linear operator $$\bar {\phi }$$ on $$V/U$$ such that

\begin{equation*} \bar {\phi }(q(v))=q(\phi (v)), \end{equation*}

for all $$v\in V$$.

• (ii) If $$\dim V=n$$, $$u_1,\dots ,u_k$$ is a basis of $$U$$ and $$q(v_{1}),\dots ,q(v_{n-k})$$ is a basis of $$V/U$$, show that $$u_1,\dots ,u_k,v_1,\dots ,v_{n-k}$$ is a basis of $$V$$.

[12]

• 8. Let $$A$$ be given by

\begin{equation*} A= \begin{pmatrix*}[r] 1&-1&2\\-3&-1&-3\\1&1&0 \end {pmatrix*}. \end{equation*}

The characteristic polynomial of $$A$$ is $$(2-x)(x+1)^{2}$$ (you do not need to prove this).

• (a) Compute the minimum polynomial of $$A$$. [5]

• (b) Find the Jordan normal form of $$A$$. [4]

• (c) Find a Jordan basis for $$A$$. [9]

• 9.

• (a) Let $$q:\R ^5\to \R$$ be a quadratic form. Which of the following are possible signatures of $$q$$?

• (i) $$(4,2)$$.

• (ii) $$(2,-1)$$.

• (iii) $$(2,1)$$.

• (iv) $$(2,3)$$.

• (b) Let $$t\in \R$$ and define $$A_t$$ by
Find an invertible matrix $$P_t$$ such that $$P_t^TA_tP_t$$ is diagonal. [10]