91Ó°ÊÓ

Let \(u=(1,2,3)\) \(\mathbf{v}=(2,2,-1),\) and \(w=(4,0,-4)\). Find \(\mathbf{z},\) where \(2 \mathbf{u}+\mathbf{v}-\mathbf{w}+3 \mathbf{z}=\mathbf{0}\)

determine whether the set, together with the standard operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. The set \(\left\\{\left(x, \frac{1}{2} x\right): x \text { is a real number }\right\\}\)

Determine whether the subset of \(C(-\infty, \infty)\) is a subspace of \(C(-\infty, \infty)\) with the standard operations. Justify your answer. The set of all odd functions: \(f(-x)=-f(x)\)

Find the Wronskian for the set of functions. $$ \left\\{x, x^{2}, e^{x}, e^{-x}\right\\} $$

Finding a Basis for a Column Space and Rank In Exercises \(21-26,\) find \((a)\) a basis for the column space and (b) the rank of the matrix. $$ \left[\begin{array}{rrr} 4 & 20 & 31 \\ 6 & -5 & -6 \\ 2 & -11 & -16 \end{array}\right] $$

In Exercises \(21-26,\) find (a) a basis for the column space and (b) the rank of the matrix. \(\left[\begin{array}{rrrr}2 & 4 & -3 & -6 \\ 7 & 14 & -6 & -3 \\ -2 & -4 & 1 & -2 \\ 2 & 4 & -2 & -2\end{array}\right]\)

Determine whether the subset of \(C(-\infty, \infty)\) is a subspace of \(C(-\infty, \infty)\) with the standard operations. Justify your answer. The set of all constant functions: \(f(x)=c\)

Determine whether the set \(S=\left\\{1, x^{2}, 2+x^{2}\right\\}\) spans \(P_{2}\).

Find the Wronskian for the set of functions. $$ \left\\{1, x, \cos x, e^{-x}\right\\} $$

determine whether the set, together with the standard operations, is a vector space. If it is not, identify at least one of the ten vector space axioms that fails. The set of all \(2 \times 2\) matrices of the form \(\left[\begin{array}{ll}a & b \\ c & 0\end{array}\right]\)