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10.25 Vector spaces

Workshop

  1. Let \(V = \mathbb{R}^2\) and \(F = \mathbb{R}\). Consider the following operations on \(V\):

    \((x, y) \oplus (w, z) = (x + w + 1, y + z + 1)\) \(a \odot (x, y) = (ax, ay)\)

    Which of the axioms for a vector space are satisfied by \((V, \oplus, \odot)\)?

    The sum of associativity and commutativity

    The neutral element \((-1,-1)\)

    The additive inverse \((-x-1,-y-1)\) of \((x,y)\)

    \(\lambda\odot\left(\mu\odot(x,y)\right)=\left(\lambda\odot\mu\right)\odot(x,y)\)

  2. Let \(X\) be a set and \(F\) a field. Show that the set \(V = \{f : X \to F : f \text{ is a function}\}\) is a vector space over \(F\) with the addition \((f + g)(x) = f(x) + g(x)\) and the scalar multiplication \((af)(x) = af(x)\).

    The sum is associative and commutative \(\checkmark\)

    There is a neutral element for that, denoted \(g(x)=0\) \(\checkmark\)

    \(\exists g(x)\in V\) s.t. \((f+g)(x)=f(x)+g(x)=0\) which is \(-f(x)\) \(\checkmark\)

    \((1\cdot f)(x) = f(x)\) \(\checkmark\)

    \(\lambda\cdot(f(x)+g(x))=\lambda\cdot f(x)+\lambda\cdot g(x)\) \(\checkmark\)

    \(\left(\lambda+\mu\right)\cdot f\left(x\right)=\lambda\cdot f\left(x\right)+\mu\cdot f\left(x\right)\) \(\checkmark\)

    \(\lambda\cdot\left(\mu\cdot f(x)\right)=\left(\lambda\cdot\mu\right)\cdot f(x)\) \(\checkmark\)

    Thus it is a vector space!

  3. Consider \(V\) a vector space over the field \(F\). Prove the following statements:

    • (a) \(-(−v) = v\) for every \(v \in V\).

    Since we are in a vector space, thus \(v+(-v)=0\).

    Then \(-v\) is the additive inverse of \(v\).

    Thus \(-(-v)=v\) * (b) \((−c) \cdot v = c \cdot (−v) = −(c \cdot v)\) for every \(v \in V\) and \(c \in F\).

    First, \(c\cdot v+(−c)\cdot v=v\cdot\left(c+\left(-c\right)\right)=v\cdot0=0\), thus \((-c)\cdot v=-(c\cdot v)\)

    Then, \(c\cdot v+c\cdot\left(-v\right)=c\cdot\left(v+\left(-v\right)\right)=c\cdot0=0\), thus \(c\cdot\left(-v)=-(c\cdot v\right)\)

    Therefore, \((−c) \cdot v = c \cdot (−v) = −(c \cdot v)\) * (c) If \(v + w = u + w\) for some \(v, w, u \in V\), then \(v = u\).

    Since \(v+w=u+w\Rightarrow\left(-w\right)+v+w=\left(-w\right)+u+w\), then \(\left(-w\right)+w+v=\left(-w\right)+w+u\Rightarrow v=u\)

  4. Consider the real vector space \(M_2(\mathbb{R})\) with the usual operations. Is \(A = \begin{pmatrix} 34 & 8 \\ 6 & -19 \end{pmatrix}\) a linear combination of \(B = \begin{pmatrix} 5 & 1 \\ 0 & -2 \end{pmatrix}\) and \(C = \begin{pmatrix} 3 & 1 \\ 2 & -3 \end{pmatrix}\)?

    Suppose \(A=xB+yC\), then we have \(\begin{cases}34=5x+3y\\ 8=x+y\\ 6=0\cdot x+2y\\ -19=-2x-3y\end{cases}\Rightarrow\begin{cases}34=5\cdot5+3\cdot3\\ x=5\\ y=3\\ -19=-2\cdot5-3\cdot3\end{cases}\).

    Thus \(A=5B+3C\) is a linear combination.