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Question
Math
Posted 2 months ago

(7) (10 points) Find the dimensions of the image and kernel of the following (4×100)(4 \times 100) matrix.
A=[12399100234100101345101102456102103]
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]
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Answer from Sia
Posted 2 months ago
Solution by Steps
step 1
Identify the matrix AA as a (4×100)(4 \times 100) matrix
step 2
Observe that the rows of matrix AA 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 AA is 1
step 4
The dimension of the image (or column space) of AA is equal to the rank of AA, which is 1
step 5
The dimension of the kernel (or null space) of AA is given by the formula: number of columnsrank\text{number of columns} - \text{rank}. Here, it is 1001=99100 - 1 = 99
Answer
The dimension of the image of matrix AA is 1, and the dimension of the kernel of matrix AA is 99.
Key Concept
Rank-Nullity Theorem
Explanation
The Rank-Nullity Theorem states that for any matrix AA, the sum of the dimensions of the image and the kernel equals the number of columns of AA. In this case, the rank of AA is 1, so the dimension of the image is 1, and the dimension of the kernel is 1001=99100 - 1 = 99.

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