91Ó°ÊÓ

Determine whether the given set of vectors spans \(\mathbb{R}^{2}\). $$\\{(1,-1),(2,-2),(2,3)\\}$$

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$V=\mathbb{R}^{2} ; B=\\{(2,-2),(1,4)\\} ; \mathbf{v}=(5,-10)$$

Let \(r\) and \(s\) denote scalars and let \(\mathbf{v}\) and \(\mathbf{w}\) denote vectors in \(\mathbb{R}^{5}\). $$\text { Prove that } r(\mathbf{v}+\mathbf{w})=r \mathbf{v}+r \mathbf{w}$$

If \(\mathbf{x}=(3,1)\) and \(\mathbf{y}=(-1,2),\) determine the vectors \(\mathbf{v}_{1}=2 \mathbf{x}, \mathbf{v}_{2}=3 \mathbf{y}, \mathbf{v}_{3}=2 \mathbf{x}+3 \mathbf{y} .\) Sketch the cor- responding points in the \(x y\) -plane and the equivalent geometric vectors.

Let \(S=\left\\{\mathbf{x} \in \mathbb{R}^{2}: \mathbf{x}=(2 k,-3 k), k \in \mathbb{R}\right\\}\) (a) Show that \(S\) is a subspace of \(\mathbb{R}^{2}\). (b) Make a sketch depicting the subspace \(S\) in the Cartesian plane.

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$V=\mathbb{R}^{2} ; B=\\{(-1,3),(3,2)\\} ; \mathbf{v}=(8,-2)$$

Determine the null space of \(A\) and verify the Rank-Nullity Theorem. $$A=\left[\begin{array}{rr}2 & -1 \\\\-4 & 2\end{array}\right]$$

If \(\mathbf{x}=(5,-2,9)\) and \(\mathbf{y}=(-1,6,4),\) determine the additive inverse of the vector \(\mathbf{v}=-2 \mathbf{x}+10 \mathbf{y}\)

determine whether the given set of vectors is linearly independent or linearly dependent in \(\mathbb{R}^{n} .\) In the case of linear dependence, find a dependency relationship. $$\\{(2,-1),(3,2),(0,1)\\}$$.

(a) find \(n\) such that rowspace \((A)\) is a subspace of \(\mathbb{R}^{n}\), and determine a basis for rowspace \((A) ;\) (b) find \(m\) such that colspace \((A)\) is a subspace of \(\mathbb{R}^{m},\) and determine a basis for colspace \((A)\). $$A=\left[\begin{array}{lllll} 1 & 2 & 3 & 4 & 5 \end{array}\right]$$