| Front | 1.2: Definition: Trivial Linear Combination |
| Back | A linear combination $\alpha_{1} \overrightarrow{v_{1}} + \alpha_{2} \overrightarrow{v_{2}} + ... + \alpha_{p} \overrightarrow{v_{p}} = \sum_{k=1}^{p}\alpha_{k} \overrightarrow{v_{k}}$ is called trvial if $\alpha_{k} = 0$ for all $k$. A trivial linear combination is always equal to $\overrightarrow{0}$. |
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| Front | 2.2.3: Definition: Echelon Form |
| Back | A matrix is in echelon form if it satis es the following two conditions:$$\text{1. All zero rows (i.e. the rows with all entries equal 0), if any, are below all non-zero entries.}$$For a non-zero row, let us call the leftmost non-zero entry the leading entry.Then the second property of the echelon form can be formulated as follows:$$\text{2. For any non-zero row its leading entry is strictly to the right of the leading entry in the previous row.}$$ |
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| Front | 2.8.1: Definition: Coordinate Vector (Trouble with LaTeX) |
| Back | Let $V$ be a vector space with a basis $B = {\overrightarrow{b_{1}}, \overrightarrow{b_{2}}, ..., \overrightarrow{b_{n}}$ Any vector $\overrightarrow{v} \in V$ admits a unique representation as a linear combination$$\overrightarrow{v} = x_{1}\overrightarrow{b_{1}} + x_{2}\overrightarrow{b_{2}} + ... + x_{n}\overrightarrow{b_{n}} = \sum_{k=1}^{n}x_{k}\overrightarrow{b_{k}}$$The numbers $x_{1}, x_{2}, ... x_{n}$ are called the coordinates of the vector $\overrightarrow{v}$ in the basis $B$. It is convenient to join these coordinates into the so-called coordinate vector of $\overrightarrow{v}$ relative to the basis $B$, which is the column vector$$[\overrightarrow{v}]_{B} = (\begin{vmatrix}x_{1} \\x_{2} \\\vdots \\x_{n}\end{vmatrx})\in \mathbb{F}^n$$ |
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