Question

Math

Posted 2 months ago

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

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Answer from Sia

Posted 2 months ago

Solution by Steps

step 1

Identify the matrix $A$ as a $(4 \times 100)$ 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: $\text{number of columns} - \text{rank}$. 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 $100 - 1 = 99$.

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