Lecture - Linear Algebra I MT22, XII

Created: November 26, 2022

Flashcards

In notation, how is the kernel of a matrix $\ker(B)$ defined where $B \in \mathbb{R}^{m \times n}$?
\[\ker(B) = \\{v \in \mathbb{R}^{n \times 1} : Bv = 0\\}\]

In notation, how is the image of a matrix $\text{Im}(B)$ defined where $B \in \mathbb{R}^{m \times n}$?
\[\text{Im}(B) = \\{ w \in \mathbb{R}^{m \times 1} : \exists v \in \mathbb{R}^{n \times 1} \text{ s.t. } Bv = w \\}\]

What is true about the row space of a matrix $\text{row}(A)$ and the row space of a matrix $\text{row}(R)$ in its RRE form?

They are equal, $\text{row}(A) = \text{row}(R)$.

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 $\text{rowrank}(A)$ and the column rank $\text{colrank}(A)$?

They are equal, $\text{rowrank(A)} = \text{colrank(A)}$.

In general, the column space of a matrix $\text{col}(A)$ is not the same as the column space of its RRE form $\text{col}(R)$. What, however, is true about the column rank?
\[\text{colrank(A)} = \text{colrank(R)}\]

Given a vector space $V$ over $\mathbb{F}$, what are 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?
\[B(v, w) = \underline{x}A\underline{y}^\intercal\]
\[B(X, Y) = \underline{x}A\underline{y}^\intercal\]

What matrix is $A$ here?

The Gram matrix of $B$ with respect to the basis being used.

Given a set of vectors $v _ 1, v _ 2, \ldots v _ k \in V$, what is the definition of the Gram matrix $A \in \mathbb{F}^{k \times k}$ of a bilinear form $B$ with respect to the vectors?
\[a_{ij} = B(v_i, v_j)\]

What does it mean for a bilinear form $B$ to be symmetric?
\[B(v_1, v_2) = B(v_2, v_1)\]

is also symmetric

What is true about the Gram matrix of a symmetric bilinear form $B$?

It is a symmetric matrix.



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