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== Four Fundamental Subspaces == | == Four Fundamental Subspaces == | ||
A matrix $A$ has four subspaces: | A matrix $A$ has four subspaces: | ||
− | * [[Column Space]] $C(A)$ | + | * [[Column Space]] $C(A)$ or $\text{ran}(A)$: Range of $A$ |
− | * [[Nullspace]] $N(A)$ | + | * [[Nullspace]] $N(A)$ or $\text{null}(A)$ |
* [[Row Space]] $C(A^T)$ of $A$ is the same as Column Space of $A^T$ | * [[Row Space]] $C(A^T)$ of $A$ is the same as Column Space of $A^T$ | ||
* Nullspace of $A^T$ (also called "Left Nullspace") | * Nullspace of $A^T$ (also called "Left Nullspace") | ||
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+ | == The Four Spaces == | ||
=== [[Column Space]] === | === [[Column Space]] === | ||
* $\text{dim } C(A) = r$, there are $r$ pivot columns | * $\text{dim } C(A) = r$, there are $r$ pivot columns | ||
* basis: columns of $A$ | * basis: columns of $A$ | ||
+ | * also called "range" | ||
+ | * $\text{ran}(A) = \{ \mathbf y \in \mathbb R^m \ : \ \mathbf y = A \mathbf x \ \forall \mathbf x \in \mathbb R^n \}$ | ||
+ | * if we think about [[Linear Transformation]] $T_{A}$ formed by $A$, this is what it does to an $n$-dimensional vector $\mathbf x$: it brings it to an $m$-dinesional vector $\mathbf y$ | ||
+ | * all such vectors $\mathbf y$ form the column space of $A$ | ||
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* $\text{dim } N(A) = n - r$ - the number of free variables | * $\text{dim } N(A) = n - r$ - the number of free variables | ||
* basis: special solutions for [[Homogeneous Systems of Linear Equations|$A\mathbf x = \mathbf 0$]] | * basis: special solutions for [[Homogeneous Systems of Linear Equations|$A\mathbf x = \mathbf 0$]] | ||
+ | * $N(A) = \text{null}(A) = \{ \mathbf x \in \mathbb R^n \ : \ A \mathbf x = \mathbf 0 \}$ | ||
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We know how to find the basis for all the subspaces | We know how to find the basis for all the subspaces | ||
* e.g. from using [[Gaussian Elimination]] transform the matrix to the echelon form and find them | * e.g. from using [[Gaussian Elimination]] transform the matrix to the echelon form and find them | ||
− | * but these bases are not "perfect". We want to use [[Orthogonal Vectors]] instead | + | * but these bases are not "perfect". We want to use [[Space Orthogonality|Orthogonal Vectors]] instead |
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* The Four Fundamental Subspaces: 4 Lines, G. Strang, [http://web.mit.edu/18.06/www/Essays/newpaper_ver3.pdf] | * The Four Fundamental Subspaces: 4 Lines, G. Strang, [http://web.mit.edu/18.06/www/Essays/newpaper_ver3.pdf] | ||
* [[Seminar Hot Topics in Information Management IMSEM (TUB)]] | * [[Seminar Hot Topics in Information Management IMSEM (TUB)]] | ||
+ | * [[Matrix Computations (book)]] | ||
[[Category:Linear Algebra]] | [[Category:Linear Algebra]] |
A matrix $A$ has four subspaces:
Suppose we have an $m \times n$ matrix of rank $r$
We know how to find the basis for all the subspaces
SVD finds these bases: