Linear_algebra

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

We look into the problem of expressing a vector $p$, whose coordinates relative to some basis is known, through the coordinates relative to a different basis. Let’s assume that $p$ lies in an $n$ dimensional vector space $V$ defined over the field $F$. Let $A$ and $B$ be two different sets of basis vectors containing the vectors $\set{\alpha_1, \ldots, \alpha_n}$ and $\set{\beta_1, \ldots, \beta_n}$ respectively. Let $p_A$ represent the coordinates of $p$ with respect to the basis $A$. [Read More]

Let $V$ be a finite dimensional vector space over the field $F$, with $\dim{V} = n$. We also define the ordered basis for $V$ to be $B = \set{\beta_1, \ldots, \beta_n}$. Consider a subspace $U$ of the vector space $V \subseteq U$. Naturally, we have $\dim{U} \leq \dim{V}$. This is because for any $\alpha \in U$, we have $\alpha \in V$, which can then be expressed as a linear combination of the basis vectors in $B$. [Read More]