Final answer:
To find the standard matrix of a linear transformation t: R², apply t to the standard basis vectors (1, 0) and (0, 1), and write the resulting vectors as columns of a matrix.
Step-by-step explanation:
A linear transformation is a function that maps vectors from one vector space to another, while preserving vector addition and scalar multiplication. In this case, the linear transformation t: R² is a mapping from the 2-dimensional Euclidean space to another vector space. To find the standard matrix of t, we need to apply t to the standard basis vectors in R², (1, 0) and (0, 1), and write the resulting vectors as columns of a matrix.
Let's consider the vector (1, 0):
t(1, 0) = (t₁,₁, t₂,₁), where t₁,₁ and t₂,₁ are the components of the result. Similarly, for the vector (0, 1):
t(0, 1) = (t₁,₂, t₂,₂)
The standard matrix of the linear transformation t can then be written as:
| t₁,₁ t₁,₂ |
| t₂,₁ t₂,₂ |