91Ó°ÊÓ

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

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. $$\\{(2,-1),(3,2),(0,1)\\}$$.

Determine whether the given set of vectors spans \(\mathbb{R}^{2}\). $$\\{(6,-2),(-2,2 / 3),(3,-1)\\}$$

If \(\mathbf{x}=(3,-1,2,5)\) and \(\mathbf{y}=(-1,2,9,-2),\) determine \(\mathbf{v}=5 \mathbf{x}+(-7) \mathbf{y}\) and its additive inverse.

Determine the null space of \(A\) and verify the Rank-Nullity Theorem. $$A=\left[\begin{array}{rrr}1 & 1 & -1 \\\3 & 4 & 4 \\\1 & 1 & 0\end{array}\right]$$

Determine whether the given set (together with the usual operations on that set) forms a vector space over \(\mathbb{R}\). In all cases, justify your answer carefully. The set of polynomials of degree 5 or less whose coefficients are even integers.

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. $$\\{(1,-1,0),(0,1,-1),(1,1,1)\\}$$.

(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 \((A)\). $$A=\left[\begin{array}{r} -3 \\ 1 \\ 7 \end{array}\right]$$

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=\mathbb{R}^{2},\) and \(S\) is the set of all vectors \((x, y)\) in \(V\) satisfying \(3 x+2 y=0\).

Determine the component vector of the given vector in the vector space \(V\) relative to the given ordered basis \(B\). $$\begin{array}{l} V=\mathbb{R}^{3} ; B=\\{(1,0,1),(1,1,-1),(2,0,1)\\} \\ \mathbf{v}=(-9,1,-8) \end{array}$$