Math  /  Algebra

QuestionSuppose T:R2R3T: R^{2} \rightarrow \mathbb{R}^{3} is a linear transformation. Three vectors u1,u2\mathbf{u}_{1}, \mathbf{u}_{2} and u3\mathbf{u}_{3} are given below together with their images by TT. Find T(w)T(\mathbf{w}) for the given vector w\mathbf{w}. u1=[22]u2=[66]u3=[23]w=[1013]T(u1)=[20816]T(u2)=[602448]T(u3)=[251121]T(w)=[000]\begin{array}{l} \mathbf{u}_{1}=\left[\begin{array}{l} 2 \\ 2 \end{array}\right] \mathbf{u}_{2}=\left[\begin{array}{l} -6 \\ -6 \end{array}\right] \mathbf{u}_{3}=\left[\begin{array}{l} 2 \\ 3 \end{array}\right] \mathbf{w}=\left[\begin{array}{l} -10 \\ -13 \end{array}\right] \quad T\left(\mathbf{u}_{1}\right)=\left[\begin{array}{c} 20 \\ 8 \\ -16 \end{array}\right] \quad T\left(\mathbf{u}_{2}\right)=\left[\begin{array}{c} -60 \\ -24 \\ 48 \end{array}\right] \quad T\left(\mathbf{u}_{3}\right)=\left[\begin{array}{c} 25 \\ 11 \\ -21 \end{array}\right] \\ T(\mathbf{w})=\left[\begin{array}{l} 0 \\ 0 \\ 0 \end{array}\right] \end{array} SUBMIT AND MARK

Studdy Solution
Perform the scalar multiplications and addition:
=[401632]+[753363]= \begin{bmatrix} -40 \\ -16 \\ 32 \end{bmatrix} + \begin{bmatrix} -75 \\ -33 \\ 63 \end{bmatrix}
Add the vectors:
=[1154995]= \begin{bmatrix} -115 \\ -49 \\ 95 \end{bmatrix}
Thus, T(w)=[1154995] T(\mathbf{w}) = \begin{bmatrix} -115 \\ -49 \\ 95 \end{bmatrix} .
The solution is:
T(w)=[1154995] T(\mathbf{w}) = \begin{bmatrix} -115 \\ -49 \\ 95 \end{bmatrix}

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