91Ó°ÊÓ

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=M_{2}(\mathrm{R}),\) and \(S\) is the subset of all \(2 \times 2\) matrices with \(\operatorname{det}(A)=1\).

Verify the associative law of addition for vectors in \(\mathbb{R}^{4}\).

Let $$ V=\left\\{\left(a_{1}, a_{2}\right): a_{1}, a_{2} \in \mathbb{R}, a_{2}>0\right\\} $$ Define addition and scalar multiplication on \(V\) as follows: $$\begin{aligned} \left(a_{1}, a_{2}\right) & \oplus\left(b_{1}, b_{2}\right)=\left(a_{1}+b_{1}, a_{2} b_{2}\right) \\ k & \otimes\left(a_{1}, a_{2}\right)=\left(k a_{1}, a_{2}^{k}\right), \quad k \in \mathbb{R} \end{aligned}$$ Explicitly verify that \(V\) is a vector space over \(\mathbb{R}\).

Determine whether the given set \(S\) of vectors is closed under addition and closed under scalar multiplication. In each case, take the set of scalars to be the set of all real numbers. The set \(S\) of all polynomials of degree exactly 2.

Determine all values of the constant \(k\) for which the vectors \((1,1, k),(0,2, k)\) and \((1, k, 6)\) are linearly dependent in \(\mathbb{R}^{3}\).

Determine the zero vector in the vector space \(V=\) \(M_{4 \times 2}(\mathrm{R}),\) and write down a general element \(A\) in \(V\) along with its additive inverse \(-A\).

Find the change-of-basis matrix \(P_{C \leftarrow B}\) from the given ordered basis \(B\) to the given ordered basis \(C\) of the vector space \(V.\) $$\begin{array}{l}V=\mathbb{R}^{3} ; B=\\{(2,-5,0),(3,0,5),(8,-2,-9)\\} \\\C=\\{(1,-1,1),(2,0,1),(0,1,3)\\} \end{array}.$$

Express \(S\) in set notation and determine whether it is a subspace of the given vector space \(V\). \(V=P_{2}(\mathbb{R}),\) and \(S\) is the subset of \(P_{2}(\mathbb{R})\) consisting of all polynomials of the form \(p(x)=a x^{2}+b\).

Prove that if \(A\) and \(B\) are \(n \times n\) matrices and \(A\) is invertible, then nullity \((A B)=\) nullity \((B)=\) nullity \((B A)\) [Hint: \(B \mathbf{x}=\mathbf{0} \text { if and only if } A B \mathbf{x}=\mathbf{0 .}]\)

For Problems 19-23, find the dimension of the null space of the given matrix \(A\). \(A=\left[\begin{array}{lll}0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 1 & 0\end{array}\right]\)