# MATH 323 Lecture 14

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## Conversion of Bases

Coordinates of ${\displaystyle {\vec {x}}}$ w.r.t. basis ${\displaystyle E=[v_{1},\ldots ,v_{n}]}$ are written as ${\displaystyle [{\vec {x}}]_{E}}$.

Conversion from basis ${\displaystyle E}$ to basis ${\displaystyle F}$ takes on following notation:

${\displaystyle [{\vec {x}}]_{F}=S[{\vec {x}}]_{E}}$

For ${\displaystyle E=[{\vec {e}}_{1},\ldots ,{\vec {e}}_{n}]}$ (standard basis), ${\displaystyle V=[{\vec {v}}_{1},\ldots ,{\vec {v}}_{n}]}$, and ${\displaystyle U=[{\vec {u}}_{1},\ldots ,{\vec {u}}_{n}]}$

Transformation matrix ${\displaystyle S}$ takes following values:

1. ${\displaystyle V\to E:V}$
2. ${\displaystyle V\to U:U^{-1}V}$
3. ${\displaystyle E\to U:U^{-1}}$

## Row Space and Column Space

Span of row vectors or column vectors of matrix.

Theorem 3.6.1: Two row equivalent matrices have the same row space.

### Rank of a Matrix

${\displaystyle \mathrm {rank} (A)=\dim({\mbox{row space of }}A)}$

#### Example

${\displaystyle A={\begin{bmatrix}1&-2&3\\2&-5&1\\1&-4&-7\end{bmatrix}}\longrightarrow {\begin{bmatrix}1&-2&3\\0&1&5\\0&0&0\end{bmatrix}}}$

• the rank of A is 2
• the vectors ${\displaystyle \left\langle 1,-2,3\right\rangle }$ and ${\displaystyle \left\langle 0,1,5\right\rangle }$ are linearly independent
• they form a basis for the row span of ${\displaystyle A}$

### Solving Linear System of Equations

Theorem 3.6.2: The system ${\displaystyle A{\vec {x}}={\vec {b}}}$ has a solution iff ${\displaystyle {\vec {b}}}$ is contained in the column space of ${\displaystyle A}$.

### Theorem 3.6.3

Let ${\displaystyle A}$ be a ${\displaystyle m\times n}$ matrix.

• The linear system ${\displaystyle A{\vec {x}}={\vec {b}}}$ is consistent for every ${\displaystyle {\vec {b}}\in \mathbb {R} ^{n}}$ iff the column vectors of ${\displaystyle A}$ span ${\displaystyle \mathbb {R} ^{n}}$.
• The system ${\displaystyle A{\vec {x}}={\vec {b}}}$ has at moste one solution for every ${\displaystyle {\vec {b}}\in \mathbb {R} ^{n}}$ iff the column vectors of ${\displaystyle A}$ are linearly independent.