Line 9: | Line 9: | ||

* suppose we have a vector $v_1$ and a vector $v_2 = c \cdot v_1$ | * suppose we have a vector $v_1$ and a vector $v_2 = c \cdot v_1$ | ||

* http://habrastorage.org/files/807/633/b50/807633b501c745a595e6a0a12277cedb.png | * http://habrastorage.org/files/807/633/b50/807633b501c745a595e6a0a12277cedb.png | ||

− | * this system is dependent | + | * this system is dependent: $v_2$ points in the same direction as $v_1$ |

Zero vector always means a dependence | Zero vector always means a dependence | ||

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** if $r < n$, there are free variables and $| N(A) | > 1$ | ** if $r < n$, there are free variables and $| N(A) | > 1$ | ||

+ | |||

+ | == Maximal Linearly Independent Subset == | ||

+ | Suppose we have a set of vectors $A = \{ \mathbf a_i \}$ | ||

+ | * subset $A^*$ is maximal linearly independent subset of $A$ if | ||

+ | * all vectors in $A^*$ are linearly independent | ||

+ | * it's not contained in any other subset of linearly idependent vectors from $A$ | ||

== Sources == | == Sources == | ||

* [[Linear Algebra MIT 18.06 (OCW)]] | * [[Linear Algebra MIT 18.06 (OCW)]] | ||

+ | * [[Matrix Computations (book)]] | ||

[[Category:Linear Algebra]] | [[Category:Linear Algebra]] |

Vectors $\mathbf x_1, \mathbf x_2, ... , \mathbf x_n$ are *linearly independent* if no linear combinations gives a zero vector $\mathbf 0$

- $c_1 \mathbf x_1 + c_2 \mathbf x_2 + ... + c_n \mathbf x_n \ne \mathbf 0$
- only when $\forall i: c_i = 0$ it's true that $\sum c_i \mathbf x_i = 0$

Example 1

- suppose we have a vector $v_1$ and a vector $v_2 = c \cdot v_1$
- this system is dependent: $v_2$ points in the same direction as $v_1$

Zero vector always means a dependence

- suppose we have vector $v_1 \ne \mathbf 0$ and $v_2 = \mathbf 0$
- $0 \mathbf v_1 + c \mathbf v_2 = \mathbf 0$ for any $c$

Example 2

- suppose we have a system of two independent vectors $v_1$ and $v_2$ in $\mathbb R^2$
- what happens if we add 3rd vectors $v_3$?
- the system is no longer dependent - we always can express $v_3$ in terms of $v_1$ and $v_2$ and when we add them, we'll have 0
- so if the number of vectors is greater than the dimensionality of these vectors, the system cannot be independent

Columns of a matrix $A$ are independent if the Nullspace $N(A)$ contains only $\mathbf 0$

- otherwise the columns are dependents
- Why? recall that $N(A)$ contains the solutions to the system $A\mathbf x = \mathbf 0$
- so there's a combination of columns with coefficients $\mathbf x$ that is equal to $\mathbf 0$

- It's related to rank as well:
- if $r = n$, then there are no free variables and $N(A) = \{ \, \mathbf 0 \, \}$
- if $r < n$, there are free variables and $| N(A) | > 1$

Suppose we have a set of vectors $A = \{ \mathbf a_i \}$

- subset $A^*$ is maximal linearly independent subset of $A$ if
- all vectors in $A^*$ are linearly independent
- it's not contained in any other subset of linearly idependent vectors from $A$