Find a basis for $\spn{S}$, where \begin{align*} S &= \set{ \colvector{1\\3\\2\\1}, \colvector{1\\2\\1\\1}, \colvector{1\\1\\0\\1}, \colvector{1\\2\\2\\1}, \colvector{3\\4\\1\\3} }. \end{align*}

Find a basis for the subspace $W$ of $\complex{4}$, \begin{align*} W &= \setparts{\colvector{a + b - 2c\\a + b - 2c + d\\ -2a + 2b + 4c - d\\ b + d}} {a, b, c, d \in\complexes} \end{align*}

Find a basis for the vector space $T$ of lower triangular $3 \times 3$ matrices; that is, matrices of the form $\begin{bmatrix} * & 0 & 0\\ * & * & 0\\ * & * & *\end{bmatrix}$ where an asterisk represents any complex number.

Find a basis for the subspace $Q$ of $P_2$, $Q = \setparts{p(x) = a + bx + cx^2}{p(0) = 0}$.

Find a basis for the subspace $R$ of $P_2$, $R = \setparts{p(x) = a + bx + cx^2}{p'(0) = 0}$, where $p'$ denotes the derivative.

From Example RSB, form an arbitrary (and nontrivial) linear combination of the four vectors in the original spanning set for $W$. So the result of this computation is of course an element of $W$. As such, this vector should be a linear combination of the basis vectors in $B$. Find the (unique) scalars that provide this linear combination. Repeat with another linear combination of the original four vectors.

Prove that $\set{(1,\,2),\,(2,\,3)}$ is a basis for the crazy vector space $C$ (Example CVS).

In Example BM provide the verifications (linear independence and spanning) to show that $B$ is a basis of $M_{mn}$.

Theorem UMCOB says that unitary matrices are characterized as those matrices that “carry” orthonormal bases to orthonormal bases. This problem asks you to prove a similar result: nonsingular matrices are characterized as those matrices that “carry” bases to bases.

More precisely, suppose that $A$ is a square matrix of size $n$ and $B=\set{\vectorlist{x}{n}}$ is a basis of $\complex{n}$. Prove that $A$ is nonsingular if and only if $C=\set{A\vect{x}_1,\,A\vect{x}_2,\,A\vect{x}_3,\,\dots,\,A\vect{x}_n}$ is a basis of $\complex{n}$. (See also Exercise PD.T33, Exercise MR.T20.)

Use the result of Exercise B.T50 to build a very concise proof of Theorem CNMB. (Hint: make a judicious choice for the basis $B$.)