15.3 verify that dim(NS(A)) + Rank(A) = 5

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In summary, the matrix A has a basis for its row space consisting of the nonzero rows in row echelon form, and the dimension of its row space is 3. Additionally, the sum of the dimension of its null space and its rank equals the total number of columns in the matrix, 5.
  • #1
karush
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nmh{780}

15.3 For the matrix
$$A=\begin{bmatrix}
1 & 0 &0 & 4 &5\\
0 & 1 & 0 & 3 &2\\
0 & 0 & 1 & 3 &2\\
0 & 0 & 0 & 0 &0
\end{bmatrix}$$
(a)find a basis for RS(A) and dim(RS(A)).
ok I am assuming that since this is already in row echelon form, its nonzero rows form a basis for RS(A) then
So...
$$RS(A)=(1,0,0,4,5),\quad(0,1,0,3,2),\quad(0,0,1,3,2)$$
also
dim(RS(A))= ??

(b)verify that dim(NS(A)) + Rank(A) = 5.

ok I am a little unsure what this means
 
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  • #2
but I think it means that the dimension of the null space plus the rank of the matrix should equal the total number of columns in the matrix, which in this case is 5. So let's check:
dim(NS(A)) = number of free variables = 2 (since the last two columns have no pivots)
Rank(A) = number of nonzero rows = 3

So, 2 + 3 = 5. This checks out!
 

FAQ: 15.3 verify that dim(NS(A)) + Rank(A) = 5

1. What does "dim(NS(A))" mean in the equation?

"dim(NS(A))" refers to the dimension of the null space of matrix A. This is the set of all vectors that, when multiplied by A, result in the zero vector.

2. How is the rank of a matrix determined?

The rank of a matrix is determined by finding the maximum number of linearly independent rows or columns in the matrix. This can also be thought of as the number of pivot positions in the matrix after performing row reduction.

3. Why is it important to verify this equation?

This equation is important because it is a fundamental property of matrices and can be used to solve systems of linear equations. It also helps to understand the relationships between the null space and rank of a matrix.

4. Can this equation be applied to all matrices?

Yes, this equation can be applied to all matrices, as long as the matrix is square (the number of rows equals the number of columns). If the matrix is not square, the equation may not hold true.

5. What are some real-world applications of this equation?

This equation is commonly used in fields such as engineering, physics, and computer science to solve systems of linear equations. It can also be used in data analysis and machine learning to understand the relationships between variables in a dataset.

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