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 | ||
Line 34: | Line 34: | ||
** 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 $\mathbf 0$
Example 1
Zero vector always means a dependence
Example 2
Columns of a matrix $A$ are independent if the Nullspace $N(A)$ contains only $\mathbf 0$
Suppose we have a set of vectors $A = \{ \mathbf a_i \}$