91Ó°ÊÓ

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 \(\\{(x, y): x \geq 0, y \text { is a real number }\\}\)

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

Let \(u=(1,2,3)\) \(\mathbf{v}=(2,2,-1),\) and \(w=(4,0,-4)\). Find \(5 \mathbf{u}-3 \mathbf{v}-\frac{1}{2} w\)

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 \(\\{(x, y): x \geq 0, y \geq 0\\}\)

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

Determine whether the set \(S\) spans \(R^{3} .\) If the set does not span \(R^{3},\) then give a geometric description of the subspace that it does span. \(S=\\{(1,0,1),(1,1,0),(0,1,1)\\}\)

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 negative functions: \(f(x)<0\)

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

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. $$ \left[\begin{array}{lll} 1 & 2 & 3 \end{array}\right] $$

Find the transition matrix from \(B\) to \(B^{\prime}\) $$\begin{aligned}&B=\\{(1,0,0),(0,1,0),(0,0,1)\\}\\\&B^{\prime}=\\{(1,3,-1),(2,7,-4),(2,9,-7)\\}\end{aligned}$$