Linear Algebra: I Scalars and Vectors
Scalars and Vectors A scalar is defined as a real number $c \in \mathbb{R}$ with 1 dimensional length $|c|$ and either positive or negative direction on the number line. A vector , denoted $\vec{v}$, is an ordered collection of scalars with dimension $n \geq 1$: Written as a column vector $$\vec{v} =\begin{bmatrix} v_1 \\ \vdots \\ v_n \end{bmatrix}\in \mathbb{R}^n $$ Written as row vector $$\vec{v}^\top =\begin{bmatrix} v_1 & \ldots & v_n \end{bmatrix}\in \mathbb{R}^n$$ Where $\top$ is the transpose operator that exchanges column vectors with row vectors. Note $(\vec{v}^\top)^\top = \vec{v}$. With magnitude $$\|\vec{v}\| = \|\vec{v}^\top\|=\sqrt{v_1^2+\ldots+v_n^2} \geq 0$$ And direction (with the exception of $\vec{0}$ where $\|\vec{0}\|=0$ and $\vec{0}$ has no direction). Row vectors can be treated as transposes of column vectors, and the same operations apply via this identification. Linear Properties Vector...
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