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Consider this matrix transformation: [[-1,-2,-3],[2,-3,-1],[3,3,0]] What is the image of [[-2],[-3],[4]] under this transformation?

Consider this matrix transformation:\newline[−1amp;−2amp;−32amp;−3amp;−13amp;3amp;0] \left[\begin{array}{ccc} -1 & -2 & -3 \\ 2 & -3 & -1 \\ 3 & 3 & 0 \end{array}\right] \newlineWhat is the image of [−2−34] \left[\begin{array}{c}-2 \\ -3 \\ 4\end{array}\right] under this transformation?

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Full solution

Q. Consider this matrix transformation:\newline[−1−2−32−3−1330] \left[\begin{array}{ccc} -1 & -2 & -3 \\ 2 & -3 & -1 \\ 3 & 3 & 0 \end{array}\right] \newlineWhat is the image of [−2−34] \left[\begin{array}{c}-2 \\ -3 \\ 4\end{array}\right] under this transformation?
  1. Write Matrix and Vector: Write down the matrix and the vector to be transformed.\newlineMatrix A=[−1amp;−2amp;−3 2amp;−3amp;−1 3amp;3amp;0]A = \begin{bmatrix} -1 & -2 & -3 \ 2 & -3 & -1 \ 3 & 3 & 0 \end{bmatrix}\newlineVector v=[−2 −3 4]v = \begin{bmatrix} -2 \ -3 \ 4 \end{bmatrix}
  2. Perform Matrix Multiplication: Perform matrix multiplication to find the image of the vector under the transformation.\newlineTo multiply a matrix by a vector, we take the dot product of each row of the matrix with the vector. The result will be a new vector.
  3. Calculate First Component: Calculate the first component of the resulting vector.\newlineFirst row of A: [−1,−2,−3][-1, -2, -3]\newlineDot product with vv: (−1)⋅(−2)+(−2)⋅(−3)+(−3)⋅(4)=2+6−12=−4(-1)\cdot(-2) + (-2)\cdot(-3) + (-3)\cdot(4) = 2 + 6 - 12 = -4
  4. Calculate Second Component: Calculate the second component of the resulting vector.\newlineSecond row of AA: [2,−3,−1][2, -3, -1]\newlineDot product with vv: (2)⋅(−2)+(−3)⋅(−3)+(−1)⋅(4)=−4+9−4=1(2)\cdot(-2) + (-3)\cdot(-3) + (-1)\cdot(4) = -4 + 9 - 4 = 1
  5. Calculate Third Component: Calculate the third component of the resulting vector.\newlineThird row of A: [3,3,0][3, 3, 0]\newlineDot product with vv: (3)⋅(−2)+(3)⋅(−3)+(0)⋅(4)=−6−9+0=−15(3)\cdot(-2) + (3)\cdot(-3) + (0)\cdot(4) = -6 - 9 + 0 = -15
  6. Combine Results: Combine the results from steps 33, 44, and 55 to form the image vector.\newlineThe resulting image vector is [[−4],[1],[−15]][[-4],[1],[-15]].

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