7. Scalar Multiplication
Basic Linear AlgebraFor what value of the scalars $a$ and $b$, does the following result in a 0 vector?
$$
\begin{pmatrix}
1 \\ 2 \\ -1/3 \\ 4
\end{pmatrix}
+
a
\begin{pmatrix}
-9 \\ 0 \\ -1 \\ 6
\end{pmatrix}
+
b
\begin{pmatrix}
-2 \\ -1 \\ 0 \\ 1
\end{pmatrix}
$$
Submit $a + b$ as your answer in the rational form.
Solution
A scalar is a number--as opposed to a vector. A scalar can be a real number or a complex number. Multiplying a vector with a scalar $c$ results in another vector where all the components are multiplied by $c$. Fro example:
$$ c \begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix} = \begin{pmatrix} cx_1 \\ cx_2 \\ \vdots \\ cx_n \end{pmatrix}. $$
With a choice of $a = -1/3$ and $b = 2$ we have
$$
\begin{pmatrix}
1 \\ 2 \\ -1/3 \\ 4
\end{pmatrix}
-
\frac{1}{3}
\begin{pmatrix}
-9 \\ 0 \\ -1 \\ 6
\end{pmatrix}
+
2
\begin{pmatrix}
-2 \\ -1 \\ 0 \\ 1
\end{pmatrix}
$$
$$
=\begin{pmatrix}
1 \\ 2 \\ -1/3 \\ 4
\end{pmatrix}
+
\begin{pmatrix}
3 \\ 0 \\ 1/3 \\ -2
\end{pmatrix}
+
\begin{pmatrix}
-4 \\ -2 \\ 0 \\ 2
\end{pmatrix}
=
\begin{pmatrix}
0 \\ 0 \\ 0 \\ 0
\end{pmatrix}
$$
The above vector with all components set to zero is called a zero vector. In the future, whenever needed, we will explicitly distinguish between a zero vector and the scalar 0. The scalar multiplication of a zero vector $0$ by any scalar $c$ is the same zero vector, that is, $c 0 = 0$.