# 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 \(U_2\)?

property | equivalent? | result | explanation |
---|---|---|---|

Every \(v \in V\) has a | |||

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

question | answer | result | explanation |
---|---|---|---|

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*}

question | answer | result | explanation |
---|---|---|---|

\(A_1\) 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 \(A_6\) |