Math

QuestionFind the standard matrix of the linear transformation T:R3R2T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2} given T(e1)=(1,9)T(\mathbf{e}_{1})=(1,9), T(e2)=(4,6)T(\mathbf{e}_{2})=(-4,6), T(e3)=(9,8)T(\mathbf{e}_{3})=(9,-8). A=A=\square (Type an integer or decimal for each matrix element.)

Studdy Solution

STEP 1

Assumptions1. is a linear transformation from $\mathbb{R}^{3}$ to $\mathbb{R}^{}$. . The transformation of the standard basis vectors $\mathbf{e}_{1}, \mathbf{e}_{}$, and $\mathbf{e}_{3}$ under are given as (1,9)(1,9), (4,6)(-4,6), and (9,8)(9,-8) respectively.
3. The standard basis vectors e1,e\mathbf{e}_{1}, \mathbf{e}_{}, and e3\mathbf{e}_{3} are the columns of the 3×33 \times3 identity matrix.

STEP 2

The standard matrix of a linear transformation is the matrix whose columns are the images of the standard basis vectors under the transformation.So, the standard matrix AA of the transformation isgivenby is given byA = [(\mathbf{e}_{1}) \quad(\mathbf{e}_{2}) \quad(\mathbf{e}_{})]$$

STEP 3

Substitute the given images of the standard basis vectors under the transformation $$ into the formula for the standard matrix $A$.
A=[(1,9)(,6)(9,8)]A = [(1,9) \quad (-,6) \quad (9,-8)]

STEP 4

Write the matrix AA in standard form.
A=[149968]A = \begin{bmatrix}1 & -4 &9 \\9 &6 & -8 \end{bmatrix}So, the standard matrix of the linear transformation $$ is $\begin{bmatrix}1 & -4 &9 \\9 &6 & -8 \end{bmatrix}$.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord