Linear Algebra Exam #2 Flashcards

Purdue's MA 265 Exam #2 Flashcards of facts, not methods.sso

Rank

Number of pivots in REF

Nullity

Number of free variables (non-pivots): Number of columns - number of pivots

Linear Independence

If det(Matrix) $\neq$ 0: It’s Linear Independent

What happens in Transpose ($A^T$ )?

Rows become columns:

$$A = \begin{bmatrix}1&2&3\\4&5&6\end{bmatrix} \quad\Rightarrow\quad A^T = \begin{bmatrix}1&4\\2&5\\3&6\end{bmatrix}$$

What is Row(A)?

It’s the SPAN of the Rows of matrix A:
$\text{row}(A) = \text{span}\{\text{row}_1, \text{row}_2, …, \text{row}_m\}$

What is Col(A)?

It’s the SPAN of the Columns of matrix A:
$\text{col}(A)=\text{span}\{col_1,col_2,\dots,col_{m}\}$

$$\text{col}(A^T)=?$$

$$\text{row}(A)$$

Do row reducing operations change Row (A)?

No, a row reducing operation like $R_2’ = R_2 -4R_1$ is reversible, and just OFFSETS the span. Nothing in span changes.

What is a subset?

Literally any matrix, usually defined by a rule. Ex: $M_{m\times n}$ matrix of only integers:
$B_1=\begin{bmatrix}1 & 2\\ 3 & 4\end{bmatrix}$

What are the rules of a subspace?

1. Has a zero vector

2. Doesn’t allow addition

3. Doesn’t allow scaling