Definition
In linear algebra, a real number is called a scalar. A scalar can be multiplied with a vector coordinate-wise:
\[x < a_1, a_2, \dots, a_n > = < xa_1, xa_2, \dots, xa_n >\]
For an example,
\[ 5 <-2,1,4> = <-10,5,20> \]
Similarly, you can factor a constant from a vector.
\[<2, 6, -6> = 2 <1, 3, -3>\]
Vector Subtraction
Given a vector \(\overrightarrow{a} = < a_1, a_2, \dots, a_n >,\) the additive inverse of \(\overrightarrow{a}\) is
\[-\overrightarrow{a} = < -a_1, -a_2, \dots, -a_n >\]
Using the definition of vector addition, we can define vector subtraction as
\[\overrightarrow{b}-\overrightarrow{a} = \overrightarrow{b}+(-1\cdot\overrightarrow{a})\]
It is always the case that
\[\overrightarrow{a}-\overrightarrow{a} = \overrightarrow{0}\]
Calculator
Enter a scalar and a vector to find their product.
Warning: include_once(../../../snippets/math_lin_alg_keys.php): failed to open stream: No such file or directory in
/var/www/michaelbriankelly.com/digital-garden/math/bourbaki/8_scalar_multiplication.php on line
34
Warning: include_once(): Failed opening '../../../snippets/math_lin_alg_keys.php' for inclusion (include_path='.:/usr/share/php') in
/var/www/michaelbriankelly.com/digital-garden/math/bourbaki/8_scalar_multiplication.php on line
34
Fatal error: Uncaught Error: Call to undefined function makeKeyboard() in /var/www/michaelbriankelly.com/digital-garden/math/bourbaki/8_scalar_multiplication.php:35
Stack trace:
#0 /var/www/michaelbriankelly.com/digital-garden/math/lesson/index.php(118): include_once()
#1 {main}
thrown in
/var/www/michaelbriankelly.com/digital-garden/math/bourbaki/8_scalar_multiplication.php on line
35