(7) (10 points) Find the dimensions of the image and kernel of the following $(4 \times 100)$ matrix.
$$
A=\left[\begin{array}{cccccc}
1 & 2 & 3 & \cdots & 99 & 100 \
2 & 3 & 4 & \cdots & 100 & 101 \
3 & 4 & 5 & \cdots & 101 & 102 \
4 & 5 & 6 & \cdots & 102 & 103
\end{array}\right]
$$

Question

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the matrix $A$ as a $(4 imes 100)$ matrix. ‖ ‖ step 2 ⋮ Observe that the rows of matrix $A$ are linearly dependent. Specifically, each row can be expressed as a linear combination of the first row. ‖ ‖ step 3 ⋮ Since the rows are linearly dependent, the rank of the matrix $A$ is 1. ‖ ‖ step 4 ⋮ The dimension of the image (or column space) of $A$ is equal to the rank of $A$, which is 1. ‖ ‖ step 5 ⋮ The dimension of the kernel (or null space) of $A$ is given by the formula: $ ext{number of columns} - ext{rank}$. Here, it is $100 - 1 = 99$. ‖ ∻Answer∻ ⚹ The dimension of the image of matrix $A$ is 1, and the dimension of the kernel of matrix $A$ is 99. ⚹ ∻Key Concept∻ ⚹Rank-Nullity Theorem⚹ ∻Explanation∻ ⚹The Rank-Nullity Theorem states that for any matrix $A$, the sum of the dimensions of the image and the kernel equals the number of columns of $A$. In this case, the rank of $A$ is 1, so the dimension of the image is 1, and the dimension of the kernel is $100 - 1 = 99$.⚹◊What is the relationship between the dimensions of the image and kernel of a matrix?⍭ Generate me a similar question◊

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