Vector spaces and subspaces

1. Show that the following are vector spaces:
   1. The $n$-tuple space, $F^n$: Let $F$ be a Field and let $V$ be the set of all $n$-tuples $\alpha = (x_1,x_2,\ldots,x_n)$ of scalars $x_i \in F$. If $\beta = (y_1,y_2,\ldots,y_n)$ with $y_i \in F$ then their sum is defined as
      $$\alpha + \beta = (x_1+y_1,\ldots,x_n+y_n)$$
      and the product of a scalar $c$ and a vector $\alpha$ is
      $$c\alpha = (cx_1,cx_2,\ldots,cx_n)$$

   2. The space of $m \times n$ matrices, $F^{m \times n}$: under usual matrix addition and multiplication of a matrix with a scalar.
   3. The space of functions from a set to a Field: under the operations:
      $$(f+g)(s) = f(s) + g(s)$$
      and
      $$(cf)(s) = cf(s)$$

   4. The space of polynomial functions over a Field: with addition and scalar multiplication as defined above.
2. Which of the following sets of vectors $\alpha = (a_1,\ldots,a_n)$ in $\mathbb{R}^n$ are subspaces of $\mathbb{R}^n$ ($n \geq 3$)?
   1. all $\alpha$ such that $a_1 \geq 0$;
   2. all $\alpha$ such that $a_1 + 3a_2 = a_3$;
   3. all $\alpha$ such that $a_2 = a_1^2$;
   4. all $\alpha$ such that $a_1a_2 = 0$;
   5. all $\alpha$ such that $a_2$ is rational.
3. Let $V$ be the (real) vector space of all functions $f: \mathbb{R}\rightarrow \mathbb{R}$. Which of the following are subspaces of $V$?
   1. all $f$ such that $f(x^2) = f(x)^2$;
   2. all $f$ such that $f(0) = f(1)$;
   3. all $f$ such that $f(3) = 1 + f(-5)$;
   4. all $f$ such that $f(-1) = 0$;
   5. all $f$ which are continuous.
4. Let $W$ be the set of all $(x_1,x_2,x_3,x_4,x_5) \in \mathbb{R}^5$ which satisfy
   $$\begin{array}{ccccccccccc} 2x_1 & - & x_2 & + &\frac{4}{3}x_3 & -... ..._5 &= & 0 \\ 9x_1 & - & 3x_2& + &6x_3 & - & 3x_4 & - & 3x_5& = & 0 \end{array}$$
   Find a finite set of vectors which spans $W$.
5. Let $F$ be a Field and let $n$ be a positive integer $(n \geq 2)$. Let $V$ be a vector space of all $n \times n$ matrices over $F$. Which of the following set of matrices $A$ in $V$ are subspaces of $V$?
   1. all invertible $A$;
   2. all non-invertible $A$;
   3. all $A$ such that $AB = BA$, where $B$ is some fixed matrix in $V$;
   4. all $A$ such that $A^2 = A$.
6. 1. Prove that the only subspaces of $\mathbb{R}^1$ are $\mathbb{R}^1$ and the zero subspace.
   2. Prove that a subspace of $\mathbb{R}^2$ is $\mathbb{R}^2$, or the zero subspace, or consists of all scalar multiples of some fixed vector in $\mathbb{R}^2$.
   3. What can you say about the subspaces of $\mathbb{R}^3$?
7. Let ${\bf A}$ be a $m \times n$ matrix over $F$. Show that the set of all vectors ${\bf X}$ such that ${\bf AX} = {\bf0}$ is a subspace of $F^n$. This subspace is called the null space of ${\bf A}$ and its dimension in the nullity of ${\bf A}$.
8. Let ${\bf A}$ be a $m \times n$ matrix over $F$. Show that the set of all vectors spanned by the row vectors of ${\bf A}$ is a subspace of $F^n$. This subspace is called the row space of ${\bf A}$ and its dimension in the row rank of ${\bf A}$.
9. Let ${\bf A}$ be a $m \times n$ matrix over $F$. Show that the set of all vectors ${\bf Y}$ such that ${\bf AX} = {\bf Y}$ has a solution for ${\bf X}$ is a subspace of $F^m$. This subspace is called the range space of ${\bf A}$ and its dimension in the column rank of ${\bf A}$ (why?).
10. Show that for any matrix ${\bf A}$
    1. nullity + row rank = $n$
    2. nullity + column rank = $n$
    and conclude that
    row rank of ${\bf A}$ = column rank of ${\bf A}$
11. Consider the $5 \times 5$ matrix
    $${\bf A} = \left[ \begin{array}{rrrrr} 1 & 2 & 0 & 3 & 0 \\ 1 & 2... ... 0 & 1 & 4 & 0 \\ 2 & 4 & 1 &10 & 1 \\ 0 & 0 & 0 & 0 & 1 \end{array}\right]$$
    1. Find an invertible matrix ${\bf P}$ such that ${\bf PA}$ is a row-reduced echelon matrix ${\bf R}$.
    2. Find a basis for the row space $W$ of ${\bf R}$.
    3. Say which vectors $(b_1,b_2,b_3,b_4,b_5)$ are in $W$.
    4. Find the coordinate matrix of each vector $(b_1,b_2,b_3,b_4,b_5) \in W$ in the ordered basis chosen in (b).
    5. Write each vector $(b_1,b_2,b_3,b_4,b_5) \in W$ as a linear combination of the rows of ${\bf A}$.
    6. Give an explicit description of the null space of ${\bf A}$.
    7. Find a basis for the null space.
    8. For what column matrices ${\bf Y}$ does the equation ${\bf AX} = {\bf Y}$ have solutions ${\bf X}$?
    9. Explicitly find the range space of ${\bf A}$ and find a basis.
    10. Verify the relations regarding the nullity, row rank and column rank of ${\bf A}$.