Let \(I\) be an ideal in a Noetherian ring \(R\). The Rees algebra \(\mathcal R(I)\) of \(I\) is the graded subalgebra \(R[It] \cong \bigoplus_{n \geq 0} I^nt^n\) of \(R[t]\). There is a natural epimorphism from the symmetric algebra of \(I\), \(\text{Sym}(I)\), to \(\mathcal R(I)\). We say that the ideal, \(I\), is of linear type if this epimorphism is an isomorphism. The first known class of ideals of linear type were complete intersection ideals. Several other examples are known. In this paper the author gives a new class of ideals of linear type. Let \(k\) be a field, \(R\) a polynomial ring over \(k\) with variables \(\{x_{ij} \}\), and \(X\) the generic \(m \times n\) matrix \((x_{ij})\). Given two homogeneous \(R\)-ideals \(I_1\) and \(I_2\), she considers the kernel of the multiplication map from \(S = R/I_1 \otimes_k R/I_2\) to \(R/(I_1 + I_2)\). The kernel is the diagonal ideal \(D\) of the ring \(S\). The main result of the paper is that the ideal \(D\) is of linear type if \(I_1\) and \(I_2\) are the ideals of maximal minors of given submatrices of \(X\). In the cases considered in this paper, the special fiber rings of the diagonal ideals are the homogeneous coordinate rings of the join varieties. When \(D\) is of linear type, the embedded join variety is the whole space (but the converse is not true).