A matrix is a representation of a linear transformation between vector spaces . That is , given a linear transformations TVW , and a basis for both V and W , then we can get a matrix that essentially gives us a formula to compute T when we work in this basis . A tensor can mean many things , especially in physics . A basic tensor is a Multilinear Map from products of certain vector spaces to the base field . In particular , a tensor on the vector space V could be a map TVxVxVxVR , where R is the real numbers and it is linear in each component separately . This would be a 2 , 2 tensor since it has two copies of V and two copies of V , the dual space of V . If V has a basis and R has a basis then you can represent tensors as multidimensional arrays in the same way that you can represent linear transformations as matrices . But just as the linear transformation is the important thing when regarding matrices , this multilinear map is the important thing when regarding tensors . A Tensor , how you often see them in physics , is actually a family of basic tensors associated to some manifold . If you have an object M , like spacetime , a sphere , 3 D space , a torus , a cylinder etc , then at every point p in M , you can find the Tangent Space at that point which is just all possible directional derivatives at the point , denoted by TpM . You can also find the Cotangent Space , which is just the dual to the Tangent Space this is all possible total derivatives at a point , denoted TpM . When you see a tensor in physics , it typically means an assignment of a basic tensor on these tangent spaces at every point on M . A Tensor Field . That is , a tensor could be gp , where p is a point on your object , but for every p , gp itself is a multilinear map of the form gpTpMxTpMxTpMR this would be a 2 , 1 tensor , but the specific numbers can change . Now , if you have a coordinate system on M , then this gives you , for free , a basis on every TpM , which means that you can represent each gp as a multidimensional array . This is why physicists focus a lot on coordinates and change of coordinates as properties of tensors , even though you don ' t need to so much when you work with the true objects . Don ' t let the fact that we can use arrays to trick you into thinking that a tensor is somehow a generalization of a vector or a matrix . Arrays are not fundamental to vectors , linear transformations or tensors , so it ' s not the thing to focus on . Though , in some cases , there are nice functions between these objects , but it doesn ' t always happen and it ' s good to think of them as fundamentally different objects that interact in interesting ways .