Search the corpus

    Left multiplication by $A$ is the zero map only for $A = 0$

    E-HK-31-11

    Area
    Algebra
    Topics
    Linear Algebra, Linear Transformations
    Status
    draft

    Let \(V\) be the space of \(n \times 1\) matrices over \(F\) and let \(W\) be the space of \(m \times 1\) matrices over \(F\). Let \(A\) be a fixed \(m \times n\) matrix over \(F\) and let \(T\) be the linear transformation from \(V\) into \(W\) defined by \(T(X) = AX\). Prove that \(T\) is the zero transformation if and only if \(A\) is the zero matrix.

    Solution

    If \(A=0\), then clearly \(T(X)=AX=0\) for every \(X\), so \(T\) is the zero transformation.

    Conversely, suppose \(T\) is the zero transformation. Let \(e_j\) be the \(j\)th standard basis column of \(F^n\). Then \[ Ae_j=T(e_j)=0 \] for every \(j\). But \(Ae_j\) is exactly the \(j\)th column of \(A\). Hence every column of \(A\) is zero, so \(A=0\).

    Source Collections