Linear Algebra

In this post, we take a look at some characteristics of injective linear transformations. Let be a linear transformation from some vector space to another vector space , both defined over a field . being injective, means that no two vectors in can be mapped to the same vector in . We have a one-to-one mapping between the vectors in and their images in . What does the null space… [Read More]

We look into the problem of expressing a vector , whose coordinates relative to some basis is known, through the coordinates relative to a different basis. Let's assume that lies in an dimensional vector space defined over the field . Let and be two different sets of basis vectors containing the vectors and respectively. Let represent the coordinates of with respect to the basis . Now, how do we go… [Read More]

Let be a finite dimensional vector space over the field , with . We also define the ordered basis for to be . Consider a subspace of the vector space . Naturally, we have . This is because for any , we have , which can then be expressed as a linear combination of the basis vectors in . Hence, we don't require more than linearly independent vectors to represent… [Read More]