Use matrix multiplication results to verify: m(T+S, B, e)[u]_B = [(T+S)(u)]_e, and (m(T, B, e) + m(S, B, e))[u]_B = m(aT, B, e)[u]_B = [(aT)(u)]_e
Understand the Problem
The image shows a series of equations relating to linear transformations and their matrix representations with respect to different bases. The equations are about expressing the matrix of a sum of linear transformations, and the matrix of a scalar multiple of a linear transformation. The objective is likely to better understand properties of linear transformations.
Answer
$m(T+S, B, e) = m(T, B, e) + m(S, B, e)$ $m(aT, B, e) = a \cdot m(T, B, e)$
Answer for screen readers
$m(T+S, B, e) = m(T, B, e) + m(S, B, e)$
$m(aT, B, e) = a \cdot m(T, B, e)$
Steps to Solve
- Expressing the matrix of the sum of linear transformations
$m(T+S, B, e)[u]_B = [(T+S)(u)]_e$ represents the matrix of the linear transformation $T+S$ with respect to bases $B$ and $e$, applied to the coordinate vector of $u$ with respect to $B$. This is stated to be equal to the coordinate vector of $(T+S)(u)$ with respect to the basis $e$.
- Simplifying $m(T+S, B, e)[u]_B$
From the linearity of matrix multiplication, we can deduce that $m(T+S, B, e)[u]_B = [T(u) + S(u)]_e = [T(u)]_e + [S(u)]_e$. Also, $[T(u)]_e = m(T, B, e)[u]_B$ and $[S(u)]_e = m(S, B, e)[u]_B$. Therefore, $m(T+S, B, e)[u]_B = m(T, B, e)[u]_B + m(S, B, e)[u]_B = (m(T, B, e) + m(S, B, e))[u]_B$. Thus, $m(T+S, B, e) = m(T, B, e) + m(S, B, e)$.
- Expressing scalar multiplication of a linear transformation
$m(aT, B, e)[u]_B = [(aT)(u)]_e$ represents the matrix of the linear transformation $aT$ with respect to bases $B$ and $e$, applied to the coordinate vector of $u$ with respect to $B$. This is equal to the coordinate vector of $(aT)(u)$ with respect to the basis $e$.
- Simplifying $m(aT, B, e)[u]_B$
Since $(aT)(u) = a(T(u))$, we have $[(aT)(u)]_e = [a(T(u))]_e = a[T(u)]_e$. Also, $[T(u)]_e = m(T, B, e)[u]_B$. Therefore, $m(aT, B, e)[u]_B = a[T(u)]_e = a(m(T, B, e)[u]_B) = (a \cdot m(T, B, e))[u]_B$. Thus, $m(aT, B, e) = a \cdot m(T, B, e)$.
$m(T+S, B, e) = m(T, B, e) + m(S, B, e)$
$m(aT, B, e) = a \cdot m(T, B, e)$
More Information
These equations show that the matrix representation of a sum of linear transformations is the sum of their matrix representations, and the matrix representation of a scalar multiple of a linear transformation is the scalar multiple of its matrix representation.
Tips
- Forgetting that the order of transformations matters.
- Not properly applying the definition of the matrix representation of a linear transformation.
- Making mistakes with scalar multiplication and addition.
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