21-06-2024 22:15

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Tags : linear algebra Linear Equations

Linear Transformation

Definition of Linear Transformation:

A linear transformation from vector space $V$ to vector space $W$ is a mapping $T:V\to W$ such that, for all $v_1,v_2$ in $V$ and for all scalars $(c)(where(cin\mathbb{R})):$

1. ($T(v_1+v_2)=T(v_1)+T(v_2))$
2. ($T(cv)=cT(v))$

This definition is equivalent to the following: For any vectors \(v_1, v_2, ..., v_n\) in $V$ and any scalars $(c_1,c_2,...,c_n)$, where each

\ T(c_1v_1 + c_2v_2 + ... + c_nv_n) = c_1T(v_1) + c_2T(v_2) + ... + c_nT(v_n)$$ Remark: If $(T: V \to W)$ is a linear transformation, then: i. $(T(0) = 0)$ ii. $(T(-v) = -T(v))$ iii. For any vectors $(v, v')$ in $V$: $(T(n \cdot v) = n \cdot T(v)), where (n)$ is an integer. Certainly! Let’s format the content as requested without unnecessary parentheses. Here’s the extracted information from the image: --- **Definition of Linear Transformation:** A linear transformation from vector space $(V)$ to vector space (W) is a mapping (T: V \rightarrow W) such that, for all (v_1, v_2) in (V) and for all scalars (c) (where (c \in \mathbb{R})): 1. $(T(v_1 + v_2) = T(v_1) + T(v_2))$ 2. $(T(cv) = cT(v))$ This definition is equivalent to the following: For any vectors $(v_1, v_2, …, v_n)$ in $(V)$ and any scalars $(c_1, c_2, …, c_n)$, where each $$(c_i \in \mathbb{R}): [T(c_1v_1 + c_2v_2 + … + c_nv_n) = c_1T(v_1) + c_2T(v_2) + … + c_nT(v_n)]$$ **Remark:** If$$ (T: V \rightarrow W)$$ is a linear transformation, then: i. $(T(0) = 0) ii. (T(-v) = -T(v))$ ii. For any vectors $(v, v’) in (V): (T(n \cdot v) = n \cdot T(v))$, where $(n)$ is an integer. --- # References