# OSA: Linear Algebra

## Question 1

Let $$F$$ be a field and $$U_1, U_2 \subseteq V$$ two linear subspaces of the $$F$$-vector space $$V$$. Which of the following are equivalent to the assertion that $$V$$ is the direct sum of $$U_1$$ and $$U_2$$?

propertyequivalent?resultexplanation

Every $$v \in V$$ has a unique representation as a sum $$v = u_1 + u_2$$ with $$u_1 \in U_1, u_2 \in U_2$$.

Every affine subspace of $$V$$ has non-empty intersections with both $$U_1$$ and $$U_2$$.

There is a projection $$\pi:V\to V$$ such that $$\mathrm{ker}(\pi) = U_1$$ and $$\mathrm{image}(\pi) = U_2$$.

For every pair of linear maps $$\varphi_1 \in \mathrm{Hom}(U_1,V)$$, $$\varphi_2 \in \mathrm{Hom}(U_2,V)$$, there is a $$\varphi \in \mathrm{Hom}(V,V)$$ that extends both $$\varphi_1$$ and $$\varphi_2$$ in the sense that $$\varphi(u_i) = \varphi_i(u_i)$$ for $$i = 1, 2$$ and $$u_i \in U_i$$.

Every basis $$B_1$$ of $$U_1$$ can be extended to a basis $$B$$ for $$V$$ using only vectors from $$U_2$$ in $$B \setminus B_1$$.

For every basis $$B$$ of $$V$$, $$B \cap U_i$$ forms a basis of $$U_i$$ for $$i = 1,2$$.

Every union $$B = B_1 \cup B_2$$ of bases $$B_i$$ of $$U_i$$ for $$i = 1,2$$ is a basis for $$V$$.

For every $$v \in V$$ there is a $$u \in U_1$$ such that $$(v-u) \in U_2$$.

## Question 2

Consider the standard $$3$$-dimensional vector space $$V = (F_5)^3$$ over the 5-element field $$F_5$$ with addition and multiplication modulo 5. Specify

the number of points in an affine subspace of dimension 1

the number of points in a linear subspace of dimension 2

the number of points in the quotient space $$V/U$$ for a linear subspace of dimension 2

the number of affine subspaces associated to a fixed linear subspace of dimension 1

the number of distinct linear complements of a fixed linear subspace of dimension 2

## Question 3

Classify the following matrices up to similarity in $$R^{3\times 3}$$. Hint: You can use invariants such as the trace, the determinant, the characteristic polynomial and the minimal polynomial. \begin{align*} A_1 &:= \begin{pmatrix} 2 & 1 & 0\\ 0 & 4 & -2\\ 0 & 0 & 2 \end{pmatrix} & A_2 &= \begin{pmatrix} 4 & 6 & 7\\ 0 & 1 & 5\\ 0 & 0 & 3 \end{pmatrix} & A_3 &= \begin{pmatrix} 1 & 1 & 0\\ -4 & 5 & 3\\ 0 & 0 & 1 \end{pmatrix} \\ A_4 &= \begin{pmatrix} 3 & 1 & 0\\ 0 & 3 & 1\\ 0 & 0 & 1 \end{pmatrix} & A_5 &= \begin{pmatrix} 20 & -30 & -15\\ 5 & -10 & -5\\ 6 & -6 & -4 \end{pmatrix} & A_6 &= \begin{pmatrix} 2 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 2 \end{pmatrix} \end{align*}

$$A_1$$ and $$A_2$$

$$A_1$$ and $$A_3$$

$$A_1$$ and $$A_4$$

$$A_1$$ and $$A_5$$

$$A_1$$ and $$A_6$$

$$A_2$$ and $$A_3$$

$$A_2$$ and $$A_4$$

$$A_2$$ and $$A_5$$

$$A_2$$ and $$A_6$$

$$A_3$$ and $$A_4$$

$$A_3$$ and $$A_5$$

$$A_3$$ and $$A_6$$

$$A_4$$ and $$A_5$$

$$A_4$$ and $$A_6$$

$$A_5$$ and $$A_6$$