To find the linear transformation of a basis, you define where each basis vector is mapped to in the target vector space; this uniquely determines the entire linear transformation.
Understanding Linear Transformations and Bases
A linear transformation is a function between two vector spaces that preserves vector addition and scalar multiplication. A basis of a vector space is a set of linearly independent vectors that span the entire space. This means any vector in the space can be written as a unique linear combination of the basis vectors.
Determining a Linear Transformation from a Basis
The key concept is that a linear transformation is completely determined by its action on a basis. This is because any vector can be expressed as a linear combination of the basis vectors, and the linearity of the transformation allows us to find the image of that vector knowing only where the basis vectors are mapped.
Here's the process:
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Choose a basis: Let ${e_1, e_2, ..., e_n}$ be a basis for the vector space $X$.
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Define the images of the basis vectors: Choose any vectors $y_1, y_2, ..., y_n$ in the target vector space $Y$. These are the images of the basis vectors under the linear transformation, i.e., $T(e_1) = y_1, T(e_2) = y_2, ..., T(e_n) = y_n$. These choices completely define the linear transformation $T$.
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Extend linearly: For any vector $x \in X$, we can write it as a linear combination of the basis vectors: $x = c_1e_1 + c_2e_2 + ... + c_ne_n$, where $c_1, c_2, ..., c_n$ are scalars. Then, the linear transformation of $x$ is given by:
$T(x) = T(c_1e_1 + c_2e_2 + ... + c_ne_n) = c_1T(e_1) + c_2T(e_2) + ... + c_nT(e_n) = c_1y_1 + c_2y_2 + ... + c_ny_n$.
Example
Let's say we have a linear transformation $T: \mathbb{R}^2 \rightarrow \mathbb{R}^3$. We can use the standard basis for $\mathbb{R}^2$, which is ${(1, 0), (0, 1)}$.
Suppose we define:
- $T(1, 0) = (1, 2, 3)$
- $T(0, 1) = (4, 5, 6)$
Then, to find $T(2, 3)$, we express $(2, 3)$ as a linear combination of the basis vectors: $(2, 3) = 2(1, 0) + 3(0, 1)$.
Therefore, $T(2, 3) = 2T(1, 0) + 3T(0, 1) = 2(1, 2, 3) + 3(4, 5, 6) = (2, 4, 6) + (12, 15, 18) = (14, 19, 24)$.
Summary
Defining the image of each basis vector under a linear transformation completely specifies the transformation. You can then find the transformation of any vector by expressing it as a linear combination of basis vectors and applying the transformation using linearity.
