Math Problem Statement

Let f : R^3 → R^2 be the linear transformation defined by f(x, y, z) = [-1; -4] x + [-2; -7] y + [-9; -33] z. Find bases for the kernel and image of f.

Solution

The given linear transformation f:R3→R2f: \mathbb{R}^3 \rightarrow \mathbb{R}^2 is defined as follows:

f(x,y,z)=[−1−4]x+[−2−7]y+[−9−33]zf(x, y, z) = \begin{bmatrix} -1 \\ -4 \end{bmatrix} x + \begin{bmatrix} -2 \\ -7 \end{bmatrix} y + \begin{bmatrix} -9 \\ -33 \end{bmatrix} z

We can rewrite this as a matrix multiplication. Let the transformation matrix AA be:

A=[−1−2−9−4−7−33]A = \begin{bmatrix} -1 & -2 & -9 \\ -4 & -7 & -33 \end{bmatrix}

Now, we need to find:

  1. A basis for the kernel of ff (the set of vectors mapped to zero by ff).
  2. A basis for the image of ff (the span of the columns of AA).

Step 1: Find the Kernel of ff

The kernel consists of all vectors (x,y,z)(x, y, z) such that A[xyz]=[00]A \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 0 \\ 0 \end{bmatrix}. This leads to the following system of linear equations:

- x - 2y - 9z = 0 \\ -4x - 7y - 33z = 0 \end{cases}$$ Solving this system will yield the kernel of $$A$$. ### Step 2: Find the Image of $$f$$ The image of $$f$$ is the span of the columns of $$A$$: $$\text{span}\left(\begin{bmatrix} -1 \\ -4 \end{bmatrix}, \begin{bmatrix} -2 \\ -7 \end{bmatrix}, \begin{bmatrix} -9 \\ -33 \end{bmatrix}\right)$$ We can determine the linear independence of these columns to find a basis for the image. Let me compute these results for you.The results are as follows: 1. **Basis for the Kernel of $$f$$**: The kernel is trivial, meaning that the only solution to the equation $$A \mathbf{x} = 0$$ is the zero vector. Thus, the kernel of $$f$$ has dimension 0, and there is no non-trivial basis. 2. **Basis for the Image of $$f$$**: The image of $$f$$ is spanned by the following vectors: $$\left\{ \begin{bmatrix} -1 \\ -4 \end{bmatrix}, \begin{bmatrix} -2 \\ -7 \end{bmatrix} \right\}$$ This set of vectors forms a basis for the image of $$f$$. Would you like a more detailed explanation or have any questions? ### Further Questions 1. How is the rank of a matrix related to the dimensions of the kernel and image? 2. What does it mean for a linear transformation to have a trivial kernel? 3. How would we interpret the rank-nullity theorem in this context? 4. Can we find a basis for the image using different vectors? 5. How does the linear independence of columns affect the image? ### Tip Remember that the kernel provides insights into the "solutions" of the transformation, while the image describes the "reach" of the transformation in the codomain.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Linear Transformation
Kernel
Image
Basis

Formulas

f(x, y, z) = Ax, where A is the transformation matrix
Kernel: A * [x, y, z] = [0, 0]
Image: Span of the columns of A

Theorems

Rank-Nullity Theorem

Suitable Grade Level

Undergraduate Level

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