In the vector space of $2\times 2$ matrices, $M_{22}$, determine if the set $S$ below is linearly independent. \begin{equation*} S=\set{ \begin{bmatrix} 2&-1\\1&3 \end{bmatrix},\, \begin{bmatrix} 0&4\\-1&2 \end{bmatrix},\, \begin{bmatrix} 4&2\\1&3 \end{bmatrix} } \end{equation*}
In the crazy vector space $C$ (Example CVS), is the set $S=\set{(0,\,2),\ (2,\,8)}$ linearly independent?
In the vector space of polynomials $P_3$, determine if the set $S$ is linearly independent or linearly dependent. \begin{equation*} S=\set{2 +x -3x^2 -8x^3,\, 1+ x + x^2 +5x^3,\, 3 -4x^2 -7x^3} \end{equation*}
Determine if the set $S=\set{(3,\,1),\,(7,\,3)}$ is linearly independent in the crazy vector space $C$ (Example CVS).
In the vector space of real-valued functions $F = \setparts{f}{f:\mathbb{R}\rightarrow\mathbb{R}}$, determine if the following set $S$ is linearly independent. \begin{equation*} S = \set{\sin^2{x}, \cos^2{x}, 2} \end{equation*}
Let \begin{equation*} S = \set{ \begin{bmatrix} 1& 2\\2 & 1 \end{bmatrix}, \begin{bmatrix} 2 & 1\\ -1 & 2\end{bmatrix}, \begin{bmatrix} 0 & 1\\ 1 & 2\end{bmatrix} } \end{equation*}
Let \begin{equation*} S = \set{ \begin{bmatrix} 1& 2\\2 & 1 \end{bmatrix}, \begin{bmatrix} 2 & 1\\ -1 & 2\end{bmatrix}, \begin{bmatrix} 0 & 1\\ 1 & 2\end{bmatrix}, \begin{bmatrix} 1 & 0\\1 & 1 \end{bmatrix}, \begin{bmatrix} 1 & 4\\0 & 3\end{bmatrix} } \end{equation*}
In Example LIM32, find another nontrivial relation of linear dependence on the linearly dependent set of $3\times 2$ matrices, $S$.
Determine if the set $T=\set{x^2-x+5,\,4x^3-x^2+5x,\,3x+2}$ spans the vector space of polynomials with degree 4 or less, $P_4$.
The set $W$ is a subspace of $M_{22}$, the vector space of all $2\times 2$ matrices. Prove that $S$ is a spanning set for $W$. \begin{align*} W&=\setparts{\begin{bmatrix}a&b\\c&d\end{bmatrix}}{2a-3b+4c-d=0} & S=&\set{ \begin{bmatrix}1&0\\0&2\end{bmatrix},\, \begin{bmatrix}0&1\\0&-3\end{bmatrix},\, \begin{bmatrix}0&0\\1&4\end{bmatrix} } \end{align*}
Determine if the set $S=\set{(3,\,1),\,(7,\,3)}$ spans the crazy vector space $C$ (Example CVS).
Halfway through Example SSP4, we need to show that the system of equations
\begin{equation*}
\linearsystem{
\begin{bmatrix}
0 & 0 & 0 & 1\\
0 & 0 & 1 & -8\\
0 & 1 & -6 & 24\\
1 & -4 & 12 & -32\\
-2 & 4 & -8 & 16\\
\end{bmatrix}
}
{\colvector{a\\b\\c\\d\\e}}
\end{equation*}
is consistent for every choice of the vector of constants satisfying $16a+8b+4c+2d+e=0$.
Express the column space of the coefficient matrix of this system as a null space, using Theorem FS. From this use Theorem CSCS to establish that the system is always consistent. Notice that this approach removes from Example SSP4 the need to row-reduce a symbolic matrix.
Suppose that $S$ is a finite linearly independent set of vectors from the vector space $V$. Let $T$ be any subset of $S$. Prove that $T$ is linearly independent.
Prove the following variant of Theorem EMMVP that has a weaker hypothesis: Suppose that $C=\set{\vectorlist{u}{p}}$ is a linearly independent spanning set for $\complex{n}$. Suppose also that $A$ and $B$ are $m\times n$ matrices such that $A\vect{u}_i=B\vect{u}_i$ for every $1\leq i\leq n$. Then $A=B$.
Can you weaken the hypothesis even further while still preserving the conclusion?
Suppose that $V$ is a vector space and $\vect{u},\,\vect{v}\in V$ are two vectors in $V$. Use the definition of linear independence to prove that $S=\set{\vect{u},\,\vect{v}}$ is a linearly dependent set if and only if one of the two vectors is a scalar multiple of the other. Prove this directly in the context of an abstract vector space ($V$), without simply giving an upgraded version of Theorem DLDS for the special case of just two vectors.
Carefully formulate the converse of Theorem VRRB and provide a proof.