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Tracking the images of the standard basis vectors determines the destination of every other point because any arbitrary vector can be expressed as a linear combination of these basis vectors. Since a linear transformation preserves vector addition and scalar multiplication, the transformation of the arbitrary vector is exactly the same linear combination applied to the transformed basis vectors.

Conditions: The transformation is linear.; Working in a 2D plane with standard basis vectors $\hat{i} = [1, 0]^T$ and $\hat{j} = [0, 1]^T$.

Tracking the images of the standard basis vectors determines the destination of every other point because any arbitrary vector can be expressed as a linear combination of these basis vectors. Since a linear transformation preserves vector addition and scalar multiplication, the transformation of the arbitrary vector is exactly the same linear combination applied to the transformed basis vectors.

Conditions: The transformation is linear.; Working in a 2D plane with standard basis vectors $\hat{i} = [1, 0]^T$ and $\hat{j} = [0, 1]^T$.

We create a $2 \times 2$ matrix where the first column records the coordinates of where $\hat{i}$ lands, and the second column records the coordinates of where $\hat{j}$ lands. This matrix fully represents the linear transformation.

Conditions: Working in 2D space.; Knowing the images of $\hat{i}$ and $\hat{j}$.

We create a $2 \times 2$ matrix where the first column records the coordinates of where $\hat{i}$ lands, and the second column records the coordinates of where $\hat{j}$ lands. This matrix fully represents the linear transformation.

Conditions: Working in 2D space.; Knowing the images of $\hat{i}$ and $\hat{j}$.