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== Vector Spaces == | == Vector Spaces == | ||
− | Suppose we have a set $V$ and elements $\mathbf v_1, ..., \mathbf v_i ... \in V$ | + | Suppose we have a set of vectors $V$ and elements $\mathbf v_1, ..., \mathbf v_i ... \in V$ |
* we define ''addition'' on $V$ where we map any pair $\mathbf v_i, \mathbf v_j \in V$ to a value $\mathbf v_i + \mathbf v_j$ | * we define ''addition'' on $V$ where we map any pair $\mathbf v_i, \mathbf v_j \in V$ to a value $\mathbf v_i + \mathbf v_j$ | ||
* and we define the operation ''scalar multiplication'' where for any scalar number $c$ and a vector $\mathbf v \in V$ we have a value $c \cdot \mathbf v$ | * and we define the operation ''scalar multiplication'' where for any scalar number $c$ and a vector $\mathbf v \in V$ we have a value $c \cdot \mathbf v$ | ||
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multiplication on scalars ($c$'s are scalars): | multiplication on scalars ($c$'s are scalars): | ||
− | * $c (\mathbf v_1 + \mathbf v_2) = c \mathbf v_1 + c \mathbf v_2$ | + | * $c\, (\mathbf v_1 + \mathbf v_2) = c\, \mathbf v_1 + c\, \mathbf v_2$ |
− | * $(c_1 + c_2) \mathbf v = c_1 \mathbf v + c_2 \mathbf v$ | + | * $(c_1 + c_2)\, \mathbf v = c_1 \mathbf v + c_2 \mathbf v$ |
* $(c_1 \cdot c_2) \cdot \mathbf v = c_1 \cdot (c_2 \cdot \mathbf v)$ | * $(c_1 \cdot c_2) \cdot \mathbf v = c_1 \cdot (c_2 \cdot \mathbf v)$ | ||
* $1 \cdot \mathbf v = \mathbf v$ | * $1 \cdot \mathbf v = \mathbf v$ | ||
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* $c \cdot (- \mathbf v) = - c \cdot \mathbf v$ | * $c \cdot (- \mathbf v) = - c \cdot \mathbf v$ | ||
* $(- c) \cdot \mathbf v = - c \cdot \mathbf v$ | * $(- c) \cdot \mathbf v = - c \cdot \mathbf v$ | ||
− | * $c (\mathbf v_1 - \mathbf v_2) = c \mathbf v_1 - c \mathbf v_2$ | + | * $c\, (\mathbf v_1 - \mathbf v_2) = c\, \mathbf v_1 - c\, \mathbf v_2$ |
− | * $(c_1 - c_2) \mathbf v = c_1 \mathbf v - c_2 \mathbf v$ | + | * $(c_1 - c_2)\, \mathbf v = c_1 \mathbf v - c_2 \mathbf v$ |
− | + | ||
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== Example: Coordinate Spaces == | == Example: Coordinate Spaces == | ||
* $\mathbb R^2$ - real numbers ("$x/y$ plane") | * $\mathbb R^2$ - real numbers ("$x/y$ plane") | ||
− | * e.g. | + | * e.g. <math>\begin{bmatrix} |
3 \\ | 3 \\ | ||
2 | 2 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, <math>\begin{bmatrix} |
− | + | ||
0 \\ | 0 \\ | ||
0 | 0 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, <math>\begin{bmatrix} |
− | + | ||
\pi \\ | \pi \\ | ||
e | e | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, ... |
* there's a picture that goes with $\mathbb R^2$ | * there's a picture that goes with $\mathbb R^2$ | ||
* http://habrastorage.org/files/774/a1e/4ef/774a1e4efbfb4ee9996aa4a14d184659.png | * http://habrastorage.org/files/774/a1e/4ef/774a1e4efbfb4ee9996aa4a14d184659.png | ||
* so, we can picture every vector in the space | * so, we can picture every vector in the space | ||
* (same for $\mathbb R^3$) | * (same for $\mathbb R^3$) | ||
+ | |||
+ | |||
+ | == Linear Span == | ||
+ | A ''linear span'' (or just ''span'') of a set of vectors $V = \{ \mathbf v_1, ..., \mathbf v_n \}$ | ||
+ | * is a set of all linear combinations of these vectors: | ||
+ | * $\text{span}(V) = \{ \sum \beta_j \mathbf v_i \ \forall \beta_j \in \mathbb R \}$ | ||
+ | * Linear span of $V$ is a Vector Space | ||
+ | |||
+ | Unique representation | ||
+ | * if vectors of $V$ are linearly independent and $\mathbf b \in V$ | ||
+ | * then $\mathbf b$ is a unique linear combinations of vectors from $V$ | ||
+ | * i.e. $\mathbf b = \sum \beta_j \mathbf v_i$ and all $\beta_j$ are unique | ||
+ | |||
+ | |||
+ | == Basis == | ||
+ | Maximal Independent Subset | ||
+ | * if $V^*$ is maximal independent subset of $V$ (all vectors in $V$ are linearly independent and $V^*$ is not contained in any other subset of linearly independent vectors) | ||
+ | * then $\text{span}(V) = \text{span}(V^*)$ | ||
+ | * and $V^*$ is the ''basis'' for $\text{span}(V)$ | ||
+ | |||
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For a [[Matrix]] there are [[Four Fundamental Subspaces]]: | For a [[Matrix]] there are [[Four Fundamental Subspaces]]: | ||
− | * [[Column Space]] | + | * [[Column Space]] (or "range") |
* [[Row Space]] | * [[Row Space]] | ||
* [[Nullspace]] | * [[Nullspace]] | ||
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* [[Linear Algebra MIT 18.06 (OCW)]] | * [[Linear Algebra MIT 18.06 (OCW)]] | ||
* Курош А.Г. Курс Высшей Алгебры | * Курош А.Г. Курс Высшей Алгебры | ||
+ | * [[Matrix Computations (book)]] | ||
[[Category:Linear Algebra]] | [[Category:Linear Algebra]] | ||
[[Category:Vector Spaces]] | [[Category:Vector Spaces]] |
Suppose we have a set of vectors $V$ and elements $\mathbf v_1, ..., \mathbf v_i ... \in V$
So, what can we do with elements in a vector space?
The elements of $V$ are vectors and $V$ is a space if the axioms hold
multiplication on scalars ($c$'s are scalars):
A linear span (or just span) of a set of vectors $V = \{ \mathbf v_1, ..., \mathbf v_n \}$
Unique representation
Maximal Independent Subset
A subspace of a vector space should form a space on it's own.
Any line through the origin:
For a Matrix there are Four Fundamental Subspaces:
A matrix space is also a vector space, where elements are matrices of the same dimensionality: we can multiply matrices by a scalar and can add two matrices of the same dimension.
In a function space, the "vectors" are functions: