91Ó°ÊÓ

Describing the Additive Identity In Exercises \(1-6\) describe the zero vector (the additive identity) of the vector space. $$ C[-1,0] $$

Verify that \(W\) is a subspace of \(V\). In each case, assume that \(V\) has the standard operations. \(W\) is the set of all functions that are continuous on \([-1,1] . V\) is the set of all functions that are integrable on \([-1,1]\)

\(W\) is not a subspace of the vector space. Verify this by giving a specific example that violates the test for a vector subspace (Theorem 4.5). \(W\) is the set of all vectors in \(R^{2}\) whose components are integers.

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 fifth-degree polynomials

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. \(S=\\{(4,7,3),(-1,2,6),(2,-3,5)\\}\)

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 quadratic functions whose graphs pass through the origin

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

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

In Exercises \(27-40\) find the nullspace of the matrix. \(A=\left[\begin{array}{rr}2 & -1 \\ 1 & 3\end{array}\right]\)

Determine whether each vector is a scalar multiple of \(\mathbf{z}=\left(\frac{1}{2},-\frac{2}{3}, \frac{3}{4}\right)\) (a) \(\mathbf{u}=(6,-4,9)\) (b) \(\mathbf{v}=\left(-1, \frac{4}{3},-\frac{3}{2}\right)\)