T must be a linear transformation, we assume. Identify the T standard matrix. 𝟚𝟚T:R2→R2&nbsp;is a vertical shear transformation that maps&nbsp;e1&nbsp;into&nbsp;e1−2e2&nbsp;but leaves the vector&nbsp;e2&nbsp;unchanged.

T(e1)=e1−2e2andT(e2)=e2 Here, e1 and e2 are the unit vectors, as you can see in the next step T(e1)=[10]−2[01]=[1−2] and T(e2)=[01] Writing the above as a matrix yields these equation T=[10−21] According to theorem 10, the transformation matrix equals the transformation of the unit vectors, each occupying one column in this matrix. Result: T=[10−21]

Expert Answer to "T must be a linear transformation, we assume...."

College
Algebra
Linear algebra
Matrix transformations

Isa Trevino

Answered question

2021-09-28

T must be a linear transformation, we assume. Identify the T standard matrix. 𝟚𝟚 $T : R^{2} \to R^{2}$ is a vertical shear transformation that maps $e_{1} i n t o e_{1} - 2 e_{2}$ but leaves the vector $e_{2}$ unchanged.

Answer & Explanation

stuth1

Skilled2021-09-29Added 97 answers

$T (e_{1}) = e_{1} - 2 e_{2} and T (e_{2}) = e_{2}$ Here, $e_{1} and e_{2}$ are the unit vectors, as you can see in the next step
$T (e_{1}) = [\begin{matrix} 1 \\ 0 \end{matrix}] - 2 [\begin{matrix} 0 \\ 1 \end{matrix}] = [\begin{matrix} 1 \\ - 2 \end{matrix}] a n d T (e_{2}) = [\begin{matrix} 0 \\ 1 \end{matrix}]$ Writing the above as a matrix yields these equation
$T = [\begin{matrix} 1 & 0 \\ - 2 & 1 \end{matrix}]$ According to theorem 10, the transformation matrix equals the transformation of the unit vectors, each occupying one column in this matrix.
Result:
$T = [\begin{matrix} 1 & 0 \\ - 2 & 1 \end{matrix}]$

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