Section A

• 1. Define subspaces $$U,W\leq \R ^{3}$$ by $$U=\Span {(1,0,-1),(2,-1,0)}$$ and $$W=\Span {(1,1,1)}$$.

Show that the sum $$U+W$$ is direct. [4]

• 2. With $$U,W$$ as in question 1, find $$u\in U$$ and $$w\in W$$ such that

\begin{equation*} (1,2,3)=u+w. \end{equation*}

[4]

• 3. Use dot product to make $$\R ^3$$ into an inner product space.

With $$W$$ as in question 1, find a basis for $$W^{\perp }$$. [4]

• 4. Let $$V$$ be a complex inner product space and $$\phi$$ a unitary operator on $$V$$.

If $$\lambda$$ is an eigenvalue of $$\phi$$, show that $$\abs {\lambda }=1$$. [4]

• 5. Let $$V$$ be a vector space over a field $$\F$$ and $$v,w\in V$$ with $$v\neq w$$. Show that there is $$\alpha \in V^{*}$$ such that $$\alpha (v)\neq \alpha (w)$$. [4]

• 6. For $$t\in \R$$, define a quadratic form $$Q_{t}$$ on $$\R ^2$$ by $$Q_{t}(x)=x_1^2-2x_1x_2+tx^2_{2}$$.

For which $$t$$ does $$Q_t$$ have rank $$1$$? [4]

Section B

• 7. Let $$V$$ be a vector space over a field $$\F$$.

• (a) Let $$\pi _1,\pi _2,\pi _3\in L(V)$$ satisfy

\begin{align*} \pi _i\circ \pi _j&=\delta _{ij}\pi _i, \quad \text {for $1\leq i,j\leq 3$}\\ \mathrm {id}_V&=\pi _1+\pi _2+\pi _3. \end{align*}

• (i) Show that $$V=\im \pi _1\oplus \im \pi _{2}\oplus \im \pi _{3}$$.

• (ii) Show that $$\ker \pi _1=\im \pi _2\oplus \im \pi _3$$.

[12]

• (b) Let $$\phi \in L(V)$$ be a linear operator on $$V$$ and $$U\leq V$$ a $$\phi$$-invariant subspace. Let $$q:V\to V/U$$ be the quotient map.

Show that there is a well-defined linear operator $$\tilde {\phi }\in L(V/U)$$ such that

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

for $$v\in V$$. [6]

• 8.

• (a) Let $$V$$ be a finite-dimensional complex inner product space and $$\phi \in L(V)$$ a linear operator.

Show that $$\lambda$$ is an eigenvalue of $$\phi$$ if and only if $$\bar {\lambda }$$ is an eigenvalue of $$\phi ^{*}$$. [6]

• (b) Let $$U=\Span {(1,1,-1,-1),(1,0,0,-1)}\leq \R ^4$$ and view $$\R ^4$$ as an inner product space using dot product.

• (i) Compute $$U^{\perp }$$.

• (ii) Find an orthonormal basis of $$U^{\perp }$$.

• (iii) Compute the orthogonal projections of $$(1,2,3,1)$$ onto $$U$$ and $$U^{\perp }$$.

[12]

• 9.

• (a) Let $$V$$ be a finite-dimensional real vector space and $$\alpha ,\beta \in V^{*}$$ linearly independent linear functionals.

Define a symmetric bilinear form $$B$$ on $$V$$ by

\begin{equation*} B(v,w)=\tfrac 12(\alpha (v)\beta (w)+\alpha (w)\beta (v)). \end{equation*}

Compute the rank and signature of $$B$$. [6]

• (b) Let $$t\in \R$$ and define $$A_{t}$$ by

\begin{equation*} A_t= \begin{pmatrix} 1&2&0\\2&0&t\\0&t&3 \end {pmatrix}. \end{equation*}

For which $$t$$ does the symmetric bilinear form $$B_{A_t}$$ have signature $$(3,0)$$? [12]