Lecture - Linear Algebra MT22, XII
Flashcards
@Define the kernel of a matrix $\ker(B)$ where $B \in \mathbb{R}^{m \times n}$.
@Define the image of a matrix $\mathrm{Im}(B)$ where $B \in \mathbb{R}^{m \times n}$.
What is true about the row space of a matrix $\mathrm{row}(A)$ and the row space of a matrix $\mathrm{row}(R)$ in its RRE form?
They are equal, $\mathrm{row}(A) = \mathrm{row}(R)$.
What is true about the column space of a matrix $A$ and the column space of a matrix $R$ in its RRE form?
They are not necessarily equal.
What is true about the row rank of a matrix $\mathrm{rowrank}(A)$ and the column rank $\mathrm{colrank}(A)$?
They are equal, $\mathrm{rowrank}(A) = \mathrm{colrank}(A)$.
In general, the column space of a matrix $\mathrm{col}(A)$ is not the same as the column space of its RRE form $\mathrm{col}(R)$. What, however, is true about the column rank?
Given a vector space $V$ over $\mathbb{F}$, @State the two conditions for $B : V\times V \to \mathbb{F}$ to be a bilinear form.
- $B(\alpha u + \beta v, w) = \alpha B(u, w) + \beta B(v, w)$
- $B(u, \alpha v + \beta w) = \alpha B(u, v) + \beta B(u, w)$
What is any bilinear form $B(v, w)$ equivalent to for $x$ and $y$ being the coordinate representations of $v$ and $w$ with respect to some basis?
\[B(X, Y) = \underline{x}A\underline{y}^\intercal\]
What matrix is $A$ here?
The Gram matrix of $B$ with respect to the basis being used.
Given a set of vectors $v _ 1, v _ 2, \ldots v _ k \in V$, @Define the Gram matrix $A \in \mathbb{F}^{k \times k}$ of a bilinear form $B$ with respect to the vectors.
@Define what it means for a bilinear form $B$ to be symmetric.
What is true about the Gram matrix of a symmetric bilinear form $B$?
It is a symmetric matrix.