Math Problem Statement
Let A = [ [-9, -5], [-9, -7], [-3, -1] ]. Define the transformation T: R^2 → R^3 by T(x) = Ax. Find the images of u = [1, 4] and v = [a, b] under T.
Solution
The image you've uploaded shows a matrix transformation question. The matrix is:
-9 & -5 \\ -9 & -7 \\ -3 & -1 \end{bmatrix}$$ The transformation $$T : \mathbb{R}^2 \to \mathbb{R}^3$$ is defined by $$T(\mathbf{x}) = A \mathbf{x}$$. You're asked to find the images of the vectors $$\mathbf{u} = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$$ and $$\mathbf{v} = \begin{bmatrix} a \\ b \end{bmatrix}$$ under the transformation $$T$$. To solve this, we will multiply the matrix $$A$$ by each vector. ### 1. Finding $$T(\mathbf{u})$$: Multiply the matrix $$A$$ by the vector $$\mathbf{u} = \begin{bmatrix} 1 \\ 4 \end{bmatrix}$$. $$T(\mathbf{u}) = A \mathbf{u} = \begin{bmatrix} -9 & -5 \\ -9 & -7 \\ -3 & -1 \end{bmatrix} \begin{bmatrix} 1 \\ 4 \end{bmatrix} = \begin{bmatrix} -9(1) + (-5)(4) \\ -9(1) + (-7)(4) \\ -3(1) + (-1)(4) \end{bmatrix} = \begin{bmatrix} -9 - 20 \\ -9 - 28 \\ -3 - 4 \end{bmatrix} = \begin{bmatrix} -29 \\ -37 \\ -7 \end{bmatrix}$$ Thus, $$T(\mathbf{u}) = \begin{bmatrix} -29 \\ -37 \\ -7 \end{bmatrix}$$. ### 2. Finding $$T(\mathbf{v})$$: Now, for the vector $$\mathbf{v} = \begin{bmatrix} a \\ b \end{bmatrix}$$, the image under the transformation is: $$T(\mathbf{v}) = A \mathbf{v} = \begin{bmatrix} -9 & -5 \\ -9 & -7 \\ -3 & -1 \end{bmatrix} \begin{bmatrix} a \\ b \end{bmatrix} = \begin{bmatrix} -9a - 5b \\ -9a - 7b \\ -3a - b \end{bmatrix}$$ So, $$T(\mathbf{v}) = \begin{bmatrix} -9a - 5b \\ -9a - 7b \\ -3a - b \end{bmatrix}$$. Would you like me to go into more details or do you have any specific questions on the process? ### 5 Related Questions: 1. How do matrix transformations in $$\mathbb{R}^2$$ affect the geometric interpretation of vectors? 2. What happens when a matrix has more rows than columns, like in this case where the transformation maps from $$\mathbb{R}^2$$ to $$\mathbb{R}^3$$? 3. Can we find the inverse of matrix $$A$$? If so, how would we apply it to reverse the transformation? 4. How do linear transformations preserve vector addition and scalar multiplication? 5. How would the results change if the matrix $$A$$ was a rotation matrix? **Tip:** Always check if the matrix is invertible (i.e., if its determinant is non-zero) before attempting to reverse a transformation!Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Matrix Multiplication
Linear Transformations
Vector Operations
Matrix Algebra
Formulas
T(x) = Ax
Matrix multiplication
Theorems
Transformation under a matrix
Properties of linear transformations
Suitable Grade Level
Grades 9-12