Simple fundamental subspaces problem stumping me

In summary, fundamental subspaces are a set of subspaces that are essential for understanding the properties and behavior of a larger space in linear algebra. To determine them, Gaussian elimination is used to convert a matrix into row echelon form and identify pivot columns. These subspaces play a crucial role in understanding linear transformations and matrices, as well as solving systems of linear equations and finding solutions to least squares problems. They are also used in real-world applications such as data analysis, computer graphics, and engineering. Additionally, the concept of orthogonal complements is important in fundamental subspaces, where the column space and left null space, as well as the row space and null space, are orthogonal complements of each other.
  • #1
kostoglotov
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So the matrix is just a row vector

[tex]\begin{bmatrix}3 & 4 & 0\end{bmatrix}[/tex]

My problem, is that I get the nullspace as having to 2 dimensions, and the row space as having 2 dimenions, but that adds up to 4 dimensions, when it should add up to three. What simple thing am I missing?

Null space base vectors

[tex]\begin{bmatrix}-4\\3\\0\end{bmatrix},\begin{bmatrix}0\\0\\1\end{bmatrix}[/tex]

Row space base vectors

[tex]\begin{bmatrix}1\\0\\0\end{bmatrix},\begin{bmatrix}0\\1\\0\end{bmatrix}[/tex]
 
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  • #2
Taking the matrix to be the single row [itex]\begin{bmatrix}3 & 4 & 0\end{bmatrix}[/itex] then the null space is all [itex]\begin{bmatrix}x \\ y \\ z \end{bmatrix}[/itex] such that [itex]\begin{bmatrix}3 & 4 & 0\end{bmatrix}\begin{bmatrix}x \\ y \\ z \end{bmatrix}= 3x+ 4y= 0[/itex]. Yes, that is of dimension 2 having [itex]\{\begin{bmatrix}-4 \\ 3 \\ 0\end{bmatrix}, \begin{bmatrix}0 \\ 0 \\ 1 \end{bmatrix}\}[/itex] as basis. However, the row space is one dimensional, having the vector [itex]\begin{bmatrix}3 & 4 & 0 \end{bmatrix}[/itex], the single row making up the matrix, as basis.
 
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  • #3
Those two last vectors cannot be in the row space, because a linear combination of them will give the first null space vector, ## [-4 \ 3 \ 0]^T##. In fact, the row space is the space spanned by the rows of the matrix, considered as vectors. So the row space consists of all multiples of ## [3 \ 4 \ 0]^T ##.

EDIT: oops, HallsofIvy beat me to it :-)
 
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  • #4
I realize what I was missing now.

I kept thinking of the row space as all those vector that can be operated on by the matrix without giving a zero vector result. Except that's not what the rowsapce is! It's the vector/s that characterize/s the rows of the matrix. Thanks @HallsofIvy and @Geofleur!
 

1. What are fundamental subspaces?

Fundamental subspaces refer to a set of subspaces that are essential for understanding the properties and behavior of a larger space. In linear algebra, these subspaces are commonly known as the column space, row space, null space, and left null space.

2. How do you determine the fundamental subspaces of a matrix?

To determine the fundamental subspaces of a matrix, you can use the process of Gaussian elimination. This involves converting the matrix into row echelon form and identifying the pivot columns. The column space is then formed by the pivot columns, the row space by the nonzero rows of the reduced matrix, the null space by the special solutions of the reduced matrix, and the left null space by the special solutions of the transpose of the reduced matrix.

3. What is the significance of fundamental subspaces in linear algebra?

Fundamental subspaces play a crucial role in understanding the structure and properties of linear transformations and matrices. They provide insights into the dimensions and bases of the subspaces, as well as the relationships between them. They are also helpful in solving systems of linear equations and finding solutions to least squares problems.

4. Can you explain the concept of orthogonal complements in fundamental subspaces?

Orthogonal complements refer to the set of vectors that are perpendicular to a given subspace. In the context of fundamental subspaces, the column space and the left null space of a matrix are orthogonal complements of each other. Similarly, the row space and the null space are orthogonal complements. This property is useful in solving systems of linear equations and in finding the best fit solutions for least squares problems.

5. How are fundamental subspaces used in real-world applications?

Fundamental subspaces have numerous applications in fields such as data analysis, computer graphics, and engineering. For example, in data analysis, the column space and null space are used in principal component analysis and in identifying independent variables. In computer graphics, the row space and null space are used in solving problems related to rotation and translation of objects. In engineering, fundamental subspaces are used in designing control systems and in analyzing the stability of systems.

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