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Mathematics LibreTexts

6.E: Exercises for Chapter 6

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    Calculational Exercises

     

    1. Define the map \(T: \mathbb{R}^2 \to \mathbb{R}^2\) by \(T(x,y)=(x+y,x)\).

    1. Show that \(T\) is linear.
    2. Show that \(T\) is surjective.
    3. Find \(\dim\left(\text{null}\left(T\right)\right)\).
    4. Find the matrix for \(T\) with respect to the canonical basis of \(\mathbb{R}^2\).
    5. Find the matrix for \(T\) with respect to the canonical basis for the domain \(\mathbb{R}^2\) and the basis \(((1,1),(1,-1))\) for the target space \(\mathbb{R}^2\).
    6. Show that the map \(F:\mathbb{R}^2 \to \mathbb{R}^2\) given by \(F(x,y)=(x+y,x+1)\) is not linear.

    2.  Let \(T\in\mathcal{L}(\mathbb{R}^2)\) be defined by

          \[ T\begin{pmatrix} x\\ y\end{pmatrix} = \begin{pmatrix}y\\ -x\end{pmatrix},\quad \mbox{ for all } \begin{pmatrix}x\\ y\end{pmatrix}\in      \mathbb{R}^2.\]

    1. Show that \(T\) is surjective.
    2. Find \(\dim\left(\text{null}\left(T\right)\right)\).
    3. Find the matrix for \(T\) with respect to the canonical basis of \(\mathbb{R}^2\).
    4. Show that the map \(F:\mathbb{R}^2 \to \mathbb{R}^2\) given by \(F(x,y)=(x+y,x+1)\) is not linear.

    3.  Consider the complex vector spaces \(\mathbb{C}^2\) and \(\mathbb{C}^3\) with their canonical bases, and define \(S \in \mathcal{L}(\mathbb{C}^3,\mathbb{C}^2)\) be the linear map defined by \(S(v) = A v, \forall v \in \mathbb{C}^{3}\), where               \(A\) is the matrix

    \[   A = M(S) = \begin{pmatrix} i &1 &1 \\ 2i& -1& -1 \end{pmatrix} .\]

          Find a basis for \(null(S).\)

    4. Give an example of a function \(f: \mathbb{R}^{2} \to \mathbb{R}\) having

         the property that

    \[ \forall a \in \mathbb{R}, \forall  v \in \mathbb{R}^2, f(av) = a f(v) \]

         but such that \(f\) is not a linear map.

    5.  Show that the linear map \(T: \mathbb{F}^{4} \to \mathbb{F}^{2}\) is surjective if

    \[ \mbox{null}(T) = \{(x_{1}, x_{2}, x_{3}, x_{4}) \in \mathbb{F}^{4} \ | \ x_{1} = 5 x_{2}, x_{3} = 7 x_{4} \}. \]

    6. Show that no linear map \(T: \mathbb{F}^{5} \to \mathbb{F}^{2}\) can

         have as its null space the set

    \[ \{(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}) \in \mathbb{F}^{5} \ | \ x_{1} = 3 x_{2}, x_{3} = x_{4} = x_{5} \}. \] 

    7.  Describe the set of solutions \(x=(x_1,x_2,x_3)\in\mathbb{R}^3\) of the system of equations

    \[ \left. \begin{array}{rl} x_1-x_2+x_3&=0 \\ x_1+2x_2 +x_3&=0 \\ 2x_1+x_2+2x_3&=0 \end{array} \right\}. \]

     

    Proof-Writing Exercises

     

     1. Let \(V\) and \(W\) be vector spaces over \(\mathbb{F}\) with \(V\) finite-dimensional, and let \(U\) be any
         subspace of \(V\) . Given a linear map \(S \in \cal{L}(U,W),\) prove that there exists a linear map
         \(T \in \cal{L}(V,W)\) such that, for every \(u \in U, S(u) = T(u).\)

    2. Let \(V\) and \(W\) be vector spaces over \(\mathbb{F},\) and suppose that \(T \in \cal{L}(V,W)\) is injective.
        Given a linearly independent list \((v_1,\ldots , v_n)\) of vectors in \(V\), prove that the
         list \((T(v_1), \ldots ,T(v_n))\) is linearly independent in \(W.\)

    3. Let \(U, V,\) and \(W\) be vector spaces over \(\mathbb{F},\) and suppose that the linear maps \(S \in \cal{L}(U, V )\)
         and \(T \in \cal{L}(V,W)\) are both injective. Prove that the composition map \(T \circ S\) is injective.

    4. Let \(V\) and \(W\) be vector spaces over \(\mathbb{F},\) and suppose that \(T \in \cal{L}(V,W)\) is surjective.
        Given a spanning list \((v_1,\ldots , v_n)\) for \(V\) , prove that


        \[span(T(v_1),\ldots ,T(v_n)) = W.\]
        

    5. Let \(V\) and \(W\) be vector spaces over \(\mathbb{F}\) with \(V\) finite-dimensional. Given \(T \in \cal{L}(V,W),\)
         prove that there is a subspace \(U\) of \(V\) such that
       

     \[U \cap null(T) = \{0\} \rm{~and~} range(T) = \{T(u) | u \in U\}.\]

     

    6. Let \(V\) be a vector space over \(\mathbb{F},\) and suppose that there is a linear map \(T \in \cal{L}(V, V )\)
         such that both \(null(T)\) and \(range(T)\) are finite-dimensional subspaces of \(V\) . Prove that
         \(V\) must also be finite-dimensional.

    7. Let \(U, V,\) and \(W\) be finite-dimensional vector spaces over \(\mathbb{F}\) with \(S \in \cal{L}(U, V )\) and
        \(T \in \cal{L}(V,W).\) Prove that
       

    \[dim(null(T \circ S)) \leq dim(null(T)) + dim(null(S)).\]

     

    8. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\) with \(S, T \in \cal{L}(V, V).\) Prove that
        \(T \circ S\) is invertible if and only if both \(S\) and \(T\) are invertible.

    9. Let \(V\) be a finite-dimensional vector space over \(\mathbb{F}\) with \(S, T \in \cal{L}(V, V ),\) and denote by
        I the identity map on \(V\) . Prove that \(T \circ S = I\) if and only if \(S \circ T = I.\)