MATH 323 Lecture 14

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Conversion of Bases

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

Conversion from basis $E$ to basis $F$ takes on following notation:

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

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

Transformation matrix $S$ takes following values:

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

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

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

Example

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

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

Solving Linear System of Equations

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

Theorem 3.6.3

Let $A$ be a $m\times n$ matrix.

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