91Ó°ÊÓ

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 even functions: \(f(-x)=f(x)\)

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. $$ \left[\begin{array}{rrr} 1 & 2 & 4 \\ -1 & 2 & 1 \end{array}\right] $$

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. \(S=\\{(1,-2,0),(0,0,1),(-1,2,0)\\}\)

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

Find the Wronskian for the set of functions. $$ \left\\{1, x, x^{2}, x^{3}\right\\} $$

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

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

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. $$ \left[\begin{array}{rrr} 4 & 20 & 31 \\ 6 & -5 & -6 \\ 2 & -11 & -16 \end{array}\right] $$

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. \(S=\\{(1,0,3),(2,0,-1),(4,0,5),(2,0,6)\\}\)

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