Consider this matrix transformation: [[-3,1,-1,-1],[1,1,3,3],[3,3,3,-1],[2,0,-1,-2]] What is the image of [[-5],[1],[1],[5]] under this transformation?

Write Matrix and Vector: Write down the matrix and the vector to be transformed. Matrix A: [[-3, 1, -1, -1], [ 1, 1, 3, 3], [ 3, 3, 3, -1], [ 2, 0, -1, -2]] Vector v: [[-5], [ 1], [ 1], [ 5]] We will multiply the matrix A by the vector v to find the image of v under the transformation.Perform Matrix Multiplication: Perform the matrix multiplication. To 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. First entry (row 1 dot v): (-3 * -5) + (1 * 1) + (-1 * 1) + (-1 * 5) = 15 + 1 - 1 - 5 = 10 Second entry (row 2 dot v): (1 * -5) + (1 * 1) + (3 * 1) + (3 * 5) = -5 + 1 + 3 + 15 = 14 Third entry (row 3 dot v): (3 * -5) + (3 * 1) + (3 * 1) + (-1 * 5) = -15 + 3 + 3 - 5 = -14 Fourth entry (row 4 dot v): (2 * -5) + (0 * 1) + (-1 * 1) + (-2 * 5) = -10 + 0 - 1 - 10 = -21 The resulting vector is: [[ 10], [ 14], [-14], [-21]]

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Consider this matrix transformation:

[[-3,1,-1,-1],[1,1,3,3],[3,3,3,-1],[2,0,-1,-2]]
What is the image of
[[-5],[1],[1],[5]] under this transformation?

Consider this matrix transformation: $\newline$ $\left[\begin{array}{cccc} -3 & 1 & -1 & -1 \\ 1 & 1 & 3 & 3 \\ 3 & 3 & 3 & -1 \\ 2 & 0 & -1 & -2 \end{array}\right]$ $\newline$ What is the image of $\left[\begin{array}{c}-5 \\ 1 \\ 1 \\ 5\end{array}\right]$ under this transformation?

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Q. Consider this matrix transformation:

\newline

\left[\begin{array}{cccc} -3 & 1 & -1 & -1 \\ 1 & 1 & 3 & 3 \\ 3 & 3 & 3 & -1 \\ 2 & 0 & -1 & -2 \end{array}\right]

\newline

What is the image of

\left[\begin{array}{c}-5 \\ 1 \\ 1 \\ 5\end{array}\right]

under this transformation?

Write Matrix and Vector: Write down the matrix and the vector to be transformed. $\newline$ Matrix $A$ : $\newline$ $\begin{bmatrix}-3 & 1 & -1 & -1 \ 1 & 1 & 3 & 3 \ 3 & 3 & 3 & -1 \ 2 & 0 & -1 & -2\end{bmatrix}$ $\newline$ Vector $v$ : $\newline$ $\begin{bmatrix}-5 \ 1 \ 1 \ 5\end{bmatrix}$ $\newline$ We will multiply the matrix $A$ by the vector $v$ to find the image of $v$ under the transformation.
Perform Matrix Multiplication: Perform the matrix multiplication. $\newline$ To 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. $\newline$ First entry (row $1$ dot v): $\newline$ $(-3 \times -5) + (1 \times 1) + (-1 \times 1) + (-1 \times 5) = 15 + 1 - 1 - 5 = 10$ $\newline$ Second entry (row $2$ dot v): $\newline$ $(1 \times -5) + (1 \times 1) + (3 \times 1) + (3 \times 5) = -5 + 1 + 3 + 15 = 14$ $\newline$ Third entry (row $3$ dot v): $\newline$ $(3 \times -5) + (3 \times 1) + (3 \times 1) + (-1 \times 5) = -15 + 3 + 3 - 5 = -14$ $\newline$ Fourth entry (row $4$ dot v): $\newline$ $(2 \times -5) + (0 \times 1) + (-1 \times 1) + (-2 \times 5) = -10 + 0 - 1 - 10 = -21$ $\newline$ The resulting vector is: $\newline$ $\begin{bmatrix} 10 \ 14 \ -14 \ -21 \end{bmatrix}$